Theorem hoidmvlelem2 39486

Description: This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
hoidmvlelem2.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem2.y	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem2.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem2.w	⊢ 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem2.a	⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem2.b	⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem2.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem2.f	⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
hoidmvlelem2.j	⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem2.k	⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.r	⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem2.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem2.g	⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem2.e	⊢ (𝜑 → 𝐸 ∈ ℝ+)
hoidmvlelem2.u	⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem2.su	⊢ (𝜑 → 𝑆 ∈ 𝑈)
hoidmvlelem2.sb	⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
hoidmvlelem2.p	⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
hoidmvlelem2.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
hoidmvlelem2.le	⊢ (𝜑 → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)))
hoidmvlelem2.O	⊢ 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
hoidmvlelem2.v	⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)
hoidmvlelem2.q	⊢ 𝑄 = inf(𝑉, ℝ, < )

Assertion

Ref	Expression
hoidmvlelem2	⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝑧,𝐴   𝐵,𝑎,𝑏,𝑘   𝑦,𝐵   𝑧,𝐵   𝐶,𝑎,𝑏,𝑗,𝑘   𝐶,𝑖,𝑗   𝑧,𝐶,𝑗   𝐷,𝑎,𝑏,𝑗,𝑘   𝐷,𝑐,𝑗,𝑘   𝐷,𝑖   𝑦,𝐷,𝑗   𝑧,𝐷   𝑧,𝐸   𝐹,𝑎,𝑏,𝑘   𝑧,𝐺   𝐻,𝑎,𝑏,𝑘   𝑧,𝐻   𝐽,𝑎,𝑏,𝑘   𝐾,𝑎,𝑏,𝑘   𝑧,𝐿   𝑖,𝑀,𝑗   𝑖,𝑂   𝑃,𝑎,𝑏,𝑘,𝑥   𝑄,𝑎,𝑏,𝑗,𝑘,𝑥   𝑄,𝑐,𝑥   𝑢,𝑄   𝑧,𝑄   𝑆,𝑎,𝑏,𝑗,𝑘,𝑥   𝑆,𝑐   𝑆,𝑖,𝑥   𝑢,𝑆   𝑧,𝑆   𝑢,𝑈   𝑥,𝑉,𝑦   𝑊,𝑎,𝑏,𝑗,𝑘,𝑥   𝑊,𝑐   𝑧,𝑊   𝑌,𝑎,𝑏,𝑗,𝑘,𝑥   𝑌,𝑐   𝑦,𝑌   𝑍,𝑐,𝑗,𝑘,𝑥   𝑖,𝑍   𝑦,𝑍   𝑧,𝑍   𝜑,𝑎,𝑏,𝑗,𝑘,𝑥   𝜑,𝑐   𝜑,𝑖   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑢)   𝐴(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐)   𝐵(𝑥,𝑢,𝑖,𝑗,𝑐)   𝐶(𝑥,𝑦,𝑢,𝑐)   𝐷(𝑥,𝑢)   𝑃(𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝑄(𝑦,𝑖)   𝑆(𝑦)   𝑈(𝑥,𝑦,𝑧,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐸(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐹(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐺(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐻(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐)   𝐽(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐾(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐿(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑀(𝑥,𝑦,𝑧,𝑢,𝑘,𝑎,𝑏,𝑐)   𝑂(𝑥,𝑦,𝑧,𝑢,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑉(𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑊(𝑦,𝑢,𝑖)   𝑋(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑧,𝑢,𝑖)   𝑍(𝑢,𝑎,𝑏)

Proof of Theorem hoidmvlelem2

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoidmvlelem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
2		hoidmvlelem2.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
3		snidg 4153	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
5		elun2 3743	. . . . . . . . 9 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
7		hoidmvlelem2.w	. . . . . . . 8 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
8	6, 7	syl6eleqr 2699	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
9	1, 8	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
10		hoidmvlelem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
11	10, 8	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
12		hoidmvlelem2.v	. . . . . . . 8 ⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)
13	11	snssd 4281	. . . . . . . . 9 ⊢ (𝜑 → {(𝐵‘𝑍)} ⊆ ℝ)
14		hoidmvlelem2.O	. . . . . . . . . 10 ⊢ 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
15		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
16		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) = (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
17		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝜑)
18		fz1ssnn 12243	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ⊆ ℕ
19		elrabi 3328	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑖 ∈ (1...𝑀))
20	18, 19	sseldi 3566	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑖 ∈ ℕ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑖 ∈ ℕ)
22		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
23	22	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
24		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
25	24	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
26	25	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (((𝐷‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐷‘𝑖)‘𝑍) ∈ ℝ))
27	23, 26	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)))
28		hoidmvlelem2.d	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
29	28	ffvelrnda 6267	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
30		elmapi 7765	. . . . . . . . . . . . . . 15 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
32	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ 𝑊)
33	31, 32	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
34	27, 33	chvarv 2251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
35	17, 21, 34	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
36	15, 16, 35	rnmptssd 38380	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ⊆ ℝ)
37	14, 36	syl5eqss 3612	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ⊆ ℝ)
38	13, 37	unssd 3751	. . . . . . . 8 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ⊆ ℝ)
39	12, 38	syl5eqss 3612	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ ℝ)
40		hoidmvlelem2.q	. . . . . . . 8 ⊢ 𝑄 = inf(𝑉, ℝ, < )
41		ltso 9997	. . . . . . . . . 10 ⊢ < Or ℝ
42	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → < Or ℝ)
43		snfi 7923	. . . . . . . . . . . 12 ⊢ {(𝐵‘𝑍)} ∈ Fin
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝐵‘𝑍)} ∈ Fin)
45		fzfi 12633	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
46		rabfi 8070	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∈ Fin → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin)
47	45, 46	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin
48	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin)
49	16	rnmptfi 38346	. . . . . . . . . . . . 13 ⊢ ({𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ∈ Fin)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ∈ Fin)
51	14, 50	syl5eqel 2692	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ Fin)
52		unfi 8112	. . . . . . . . . . 11 ⊢ (({(𝐵‘𝑍)} ∈ Fin ∧ 𝑂 ∈ Fin) → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
53	44, 51, 52	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
54	12, 53	syl5eqel 2692	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ Fin)
55		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (𝐵‘𝑍) ∈ V
56	55	snid 4155	. . . . . . . . . . . . 13 ⊢ (𝐵‘𝑍) ∈ {(𝐵‘𝑍)}
57		elun1 3742	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑍) ∈ {(𝐵‘𝑍)} → (𝐵‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
58	56, 57	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐵‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂)
59	12	eqcomi 2619	. . . . . . . . . . . 12 ⊢ ({(𝐵‘𝑍)} ∪ 𝑂) = 𝑉
60	58, 59	eleqtri 2686	. . . . . . . . . . 11 ⊢ (𝐵‘𝑍) ∈ 𝑉
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵‘𝑍) ∈ 𝑉)
62		ne0i 3880	. . . . . . . . . 10 ⊢ ((𝐵‘𝑍) ∈ 𝑉 → 𝑉 ≠ ∅)
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ ∅)
64		fiinfcl 8290	. . . . . . . . 9 ⊢ (( < Or ℝ ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ)) → inf(𝑉, ℝ, < ) ∈ 𝑉)
65	42, 54, 63, 39, 64	syl13anc 1320	. . . . . . . 8 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ∈ 𝑉)
66	40, 65	syl5eqel 2692	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑉)
67	39, 66	sseldd 3569	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
68		hoidmvlelem2.u	. . . . . . . . . . . 12 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
69		ssrab2 3650	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
70	68, 69	eqsstri 3598	. . . . . . . . . . 11 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
71	70	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
72	9, 11	iccssred 38574	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
73	71, 72	sstrd 3578	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
74		hoidmvlelem2.su	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
75	73, 74	sseldd 3569	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
76	9	rexrd 9968	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
77	11	rexrd 9968	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
78	70, 74	sseldi 3566	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
79		iccgelb 12101	. . . . . . . . 9 ⊢ (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
80	76, 77, 78, 79	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
81		hoidmvlelem2.sb	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < (𝐵‘𝑍))
83		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐵‘𝑍) → 𝑥 = (𝐵‘𝑍))
84	83	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝐵‘𝑍) → (𝐵‘𝑍) = 𝑥)
85	84	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → (𝐵‘𝑍) = 𝑥)
86	82, 85	breqtrd 4609	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
87	86	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
88		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝜑)
89		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
90	89, 12	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
91	90	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
92		elsni 4142	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {(𝐵‘𝑍)} → 𝑥 = (𝐵‘𝑍))
93	92	con3i 149	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = (𝐵‘𝑍) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
94	93	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
95		elunnel1 3716	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂) ∧ ¬ 𝑥 ∈ {(𝐵‘𝑍)}) → 𝑥 ∈ 𝑂)
96	91, 94, 95	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
97	96	adantll 746	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
98		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ 𝑂)
99	98, 14	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)))
100		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
101	16	elrnmpt 5293	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍)))
102	100, 101	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
103	99, 102	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑂 → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
104	103	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
105		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
106	105	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
107	106	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐶‘𝑖)‘𝑍) ∈ ℝ))
108	23, 107	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ)))
109		hoidmvlelem2.c	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
110	109	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
111		elmapi 7765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
112	110, 111	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
113	112, 32	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
114	108, 113	chvarv 2251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ)
115	114	rexrd 9968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ*)
116	17, 21, 115	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ*)
117	34	rexrd 9968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ*)
118	17, 21, 117	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ*)
119	106, 25	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
120	119	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
121	120	elrab 3331	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↔ (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
122	121	biimpi 205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
123	122	simprd 478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
124	123	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
125		icoltub 38579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑖)‘𝑍) ∈ ℝ* ∧ ((𝐷‘𝑖)‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
126	116, 118, 124, 125	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
127	126	3adant3 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
128		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑥 = ((𝐷‘𝑖)‘𝑍))
129	128	eqcomd 2616	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝐷‘𝑖)‘𝑍) → ((𝐷‘𝑖)‘𝑍) = 𝑥)
130	129	3ad2ant3 1077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → ((𝐷‘𝑖)‘𝑍) = 𝑥)
131	127, 130	breqtrd 4609	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → 𝑆 < 𝑥)
132	131	3exp 1256	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
133	132	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
134	133	rexlimdv 3012	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥))
135	104, 134	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝑆 < 𝑥)
136	88, 97, 135	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
137	87, 136	pm2.61dan 828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑆 < 𝑥)
138	137	ralrimiva 2949	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 𝑆 < 𝑥)
139		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑥 = inf(𝑉, ℝ, < ) → (𝑆 < 𝑥 ↔ 𝑆 < inf(𝑉, ℝ, < )))
140	139	rspcva 3280	. . . . . . . . . 10 ⊢ ((inf(𝑉, ℝ, < ) ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 𝑆 < 𝑥) → 𝑆 < inf(𝑉, ℝ, < ))
141	65, 138, 140	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 < inf(𝑉, ℝ, < ))
142	40	eqcomi 2619	. . . . . . . . . 10 ⊢ inf(𝑉, ℝ, < ) = 𝑄
143	142	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → inf(𝑉, ℝ, < ) = 𝑄)
144	141, 143	breqtrd 4609	. . . . . . . 8 ⊢ (𝜑 → 𝑆 < 𝑄)
145	9, 75, 67, 80, 144	lelttrd 10074	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) < 𝑄)
146	9, 67, 145	ltled 10064	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑄)
147		fiminre 10851	. . . . . . . . 9 ⊢ ((𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
148	39, 54, 63, 147	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
149		lbinfle 10857	. . . . . . . 8 ⊢ ((𝑉 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ (𝐵‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
150	39, 148, 61, 149	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
151	40, 150	syl5eqbr 4618	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝐵‘𝑍))
152	9, 11, 67, 146, 151	eliccd 38573	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
153	67	recnd 9947	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
154	75	recnd 9947	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
155	9	recnd 9947	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
156	153, 154, 155	npncand 10295	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))) = (𝑄 − (𝐴‘𝑍)))
157	156	eqcomd 2616	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − (𝐴‘𝑍)) = ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))))
158	157	oveq2d 6565	. . . . . . 7 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))))
159		rge0ssre 12151	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
160		hoidmvlelem2.g	. . . . . . . . . . 11 ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
161		hoidmvlelem2.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
162		hoidmvlelem2.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin)
163		hoidmvlelem2.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
164	162, 163	ssfid 8068	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ Fin)
165		ssun1 3738	. . . . . . . . . . . . . . 15 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
166	165, 7	sseqtr4i 3601	. . . . . . . . . . . . . 14 ⊢ 𝑌 ⊆ 𝑊
167	166	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
168	1, 167	fssresd 5984	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
169	10, 167	fssresd 5984	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
170	161, 164, 168, 169	hoidmvcl 39472	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
171	160, 170	syl5eqel 2692	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
172	159, 171	sseldi 3566	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℝ)
173	172	recnd 9947	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
174	153, 154	subcld 10271	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
175	154, 155	subcld 10271	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) ∈ ℂ)
176	173, 174, 175	adddid 9943	. . . . . . 7 ⊢ (𝜑 → (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))))
177	173, 174	mulcld 9939	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ∈ ℂ)
178	173, 175	mulcld 9939	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℂ)
179	177, 178	addcomd 10117	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
180	158, 176, 179	3eqtrd 2648	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
181	67, 75	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ))
182		resubcl 10224	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑄 − 𝑆) ∈ ℝ)
183	181, 182	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℝ)
184	172, 183	jca 553	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑄 − 𝑆) ∈ ℝ))
185		remulcl 9900	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ℝ ∧ (𝑄 − 𝑆) ∈ ℝ) → (𝐺 · (𝑄 − 𝑆)) ∈ ℝ)
186	184, 185	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ∈ ℝ)
187	75, 9	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∈ ℝ ∧ (𝐴‘𝑍) ∈ ℝ))
188		resubcl 10224	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ ℝ ∧ (𝐴‘𝑍) ∈ ℝ) → (𝑆 − (𝐴‘𝑍)) ∈ ℝ)
189	187, 188	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) ∈ ℝ)
190	172, 189	jca 553	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑆 − (𝐴‘𝑍)) ∈ ℝ))
191		remulcl 9900	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ℝ ∧ (𝑆 − (𝐴‘𝑍)) ∈ ℝ) → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ)
192	190, 191	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ)
193	186, 192	jca 553	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ))
194		readdcl 9898	. . . . . . . . 9 ⊢ (((𝐺 · (𝑄 − 𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ) → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) ∈ ℝ)
195	193, 194	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) ∈ ℝ)
196	179, 195	eqeltrrd 2689	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ∈ ℝ)
197		1red 9934	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
198		hoidmvlelem2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
199	198	rpred 11748	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
200	197, 199	readdcld 9948	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
201	2	eldifbd 3553	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
202	8, 201	eldifd 3551	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
203		hoidmvlelem2.r	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
204		hoidmvlelem2.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
205	161, 164, 202, 7, 109, 28, 203, 204, 75	sge0hsphoire 39479	. . . . . . . . 9 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
206	200, 205	remulcld 9949	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
207		fzfid 12634	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
208	183	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℝ)
209		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝜑)
210		elfznn 12241	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
211	210	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
212		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
213		ovex 6577	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V
214	213	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V)
215		hoidmvlelem2.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
216	215	fvmpt2 6200	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
217	212, 214, 216	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
218	217	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
219	164	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
220	166	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
221	112, 220	fssresd 5984	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
222	221	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
223		iftrue 4042	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
224	223	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
225	224	feq1d 5943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
226	222, 225	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
227		0red 9920	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
228		hoidmvlelem2.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
229	227, 228	fmptd 6292	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
230	229	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
231		iffalse 4045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
232	231	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
233	232	feq1d 5943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
234	230, 233	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
235	226, 234	pm2.61dan 828	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
236		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
237		fvex 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘𝑗) ∈ V
238	237	resex 5363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V
239	238	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶‘𝑗) ↾ 𝑌) ∈ V)
240	162, 163	ssexd 4733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ V)
241		mptexg 6389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 ∈ V → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
242	240, 241	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
243	228, 242	syl5eqel 2692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 ∈ V)
244	239, 243	ifcld 4081	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
245	244	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
246		hoidmvlelem2.j	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
247	246	fvmpt2 6200	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
248	236, 245, 247	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
249	248	feq1d 5943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
250	235, 249	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
251	31, 220	fssresd 5984	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
252	251	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
253		iftrue 4042	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
254	253	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
255	254	feq1d 5943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
256	252, 255	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
257		iffalse 4045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
258	257	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
259	258	feq1d 5943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
260	230, 259	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
261	256, 260	pm2.61dan 828	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
262		fvex 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷‘𝑗) ∈ V
263	262	resex 5363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷‘𝑗) ↾ 𝑌) ∈ V
264	263	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐷‘𝑗) ↾ 𝑌) ∈ V)
265	264, 243	ifcld 4081	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
266	265	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
267		hoidmvlelem2.k	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
268	267	fvmpt2 6200	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
269	236, 266, 268	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
270	269	feq1d 5943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐾‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
271	261, 270	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
272	161, 219, 250, 271	hoidmvcl 39472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
273	218, 272	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
274	159, 273	sseldi 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
275	209, 211, 274	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℝ)
276	208, 275	remulcld 9949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
277	207, 276	fsumrecl 14312	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
278	200, 277	remulcld 9949	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
279	206, 278	readdcld 9948	. . . . . . 7 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) ∈ ℝ)
280	161, 164, 202, 7, 109, 28, 203, 204, 67	sge0hsphoire 39479	. . . . . . . 8 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
281	200, 280	remulcld 9949	. . . . . . 7 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) ∈ ℝ)
282	74, 68	syl6eleq 2698	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
283		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
284	283	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
285		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
286	285	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
287	286	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
288	287	mpteq2dv 4673	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
289	288	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
290	289	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
291	284, 290	breq12d 4596	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
292	291	elrab 3331	. . . . . . . . . 10 ⊢ (𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
293	282, 292	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
294	293	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
295	207, 275	fsumrecl 14312	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℝ)
296	200, 295	remulcld 9949	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) ∈ ℝ)
297		0red 9920	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
298	75, 67	posdifd 10493	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 < 𝑄 ↔ 0 < (𝑄 − 𝑆)))
299	144, 298	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝑄 − 𝑆))
300	297, 183, 299	ltled 10064	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑄 − 𝑆))
301		hoidmvlelem2.le	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)))
302	172, 296, 183, 300, 301	lemul1ad 10842	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)))
303	200	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
304	295	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℂ)
305	303, 304, 174	mulassd 9942	. . . . . . . . . 10 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))))
306	275	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℂ)
307	207, 174, 306	fsummulc1 14359	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)))
308	174	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℂ)
309	306, 308	mulcomd 9940	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − 𝑆)) = ((𝑄 − 𝑆) · (𝑃‘𝑗)))
310	309	sumeq2dv 14281	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
311	307, 310	eqtrd 2644	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
312	311	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
313	305, 312	eqtrd 2644	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
314	302, 313	breqtrd 4609	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
315	192, 186, 206, 278, 294, 314	leadd12dd 38473	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ≤ (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
316		hoidmvlelem2.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ)
317		nnsplit 38515	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
318	316, 317	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
319		uncom 3719	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) = ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀))
320	319	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) = ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)))
321	318, 320	eqtr2d 2645	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) = ℕ)
322	321	eqcomd 2616	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ = ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)))
323	322	mpteq1d 4666	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))) = (𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
324	323	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
325		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
326		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (ℤ≥‘(𝑀 + 1)) ∈ V
327	326	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ∈ V)
328		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
329	328	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ∈ V)
330		incom 3767	. . . . . . . . . . . . . . 15 ⊢ ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ≥‘(𝑀 + 1)))
331		nnuzdisj 38512	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∩ (ℤ≥‘(𝑀 + 1))) = ∅
332	330, 331	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅
333	332	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅)
334		icossicc 12131	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
335		ssid 3587	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
336		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝜑)
337	316	peano2nnd 10914	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
338		uznnssnn 11611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ ℕ → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
339	337, 338	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
340	339	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
341		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑀 + 1)))
342	340, 341	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
343		snfi 7923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
344	343	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
345		unfi 8112	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
346	164, 344, 345	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
347	7, 346	syl5eqel 2692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
348	347	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
349		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑗 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌))
350		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑐‘𝑗) = (𝑐‘𝑙))
351	350	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑙 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑙) ≤ 𝑥))
352	351, 350	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥))
353	349, 350, 352	ifbieq12d 4063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑙 → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))
354	353	cbvmptv 4678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))
355	354	mpteq2i 4669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥))))
356	355	mpteq2i 4669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))))
357	204, 356	eqtri 2632	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))))
358	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
359	357, 358, 348, 31	hsphoif 39466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
360	161, 348, 112, 359	hoidmvcl 39472	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
361	336, 342, 360	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
362	335, 361	sseldi 3566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
363	334, 362	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
364	209, 211, 360	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
365	334, 364	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
366	325, 327, 329, 333, 363, 365	sge0splitmpt 39304	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
367		nnex 10903	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
368	367	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
369	334, 360	sseldi 3566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
370	325, 368, 369, 205, 339	sge0ssrempt 39298	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
371	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑀) ⊆ ℕ)
372	325, 368, 369, 205, 371	sge0ssrempt 39298	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
373		rexadd 11937	. . . . . . . . . . . . 13 ⊢ (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ ∧ (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ) → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
374	370, 372, 373	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
375	324, 366, 374	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
376	375	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
377	376	oveq1d 6564	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
378	375, 205	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
379	378	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℂ)
380	277	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
381	303, 379, 380	adddid 9943	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
382	381	eqcomd 2616	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
383	370	recnd 9947	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
384	372	recnd 9947	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
385	383, 384, 380	addassd 9941	. . . . . . . . . . 11 ⊢ (𝜑 → (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
386	207, 364	sge0fsummpt 39283	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
387	386	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
388		ax-resscn 9872	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℂ
389	159, 388	sstri 3577	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℂ
390	389, 360	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
391	209, 211, 390	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
392	183	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℝ)
393	392, 274	remulcld 9949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
394	393	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
395	211, 394	syldan 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
396	207, 391, 395	fsumadd 14317	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
397	396	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
398	387, 397	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
399	398	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))))
400	385, 399	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝜑 → (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))))
401	400	oveq2d 6565	. . . . . . . . 9 ⊢ (𝜑 → ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))))
402	377, 382, 401	3eqtrd 2648	. . . . . . . 8 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))))
403	159, 360	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℝ)
404	403, 393	readdcld 9948	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
405	209, 211, 404	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
406	207, 405	fsumrecl 14312	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
407	370, 406	readdcld 9948	. . . . . . . . 9 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ∈ ℝ)
408		0le1 10430	. . . . . . . . . . 11 ⊢ 0 ≤ 1
409	408	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 1)
410	198	rpge0d 11752	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐸)
411	197, 199, 409, 410	addge0d 10482	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
412	67	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℝ)
413	357, 412, 348, 31	hsphoif 39466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑄)‘(𝐷‘𝑗)):𝑊⟶ℝ)
414	161, 348, 112, 413	hoidmvcl 39472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
415	334, 414	sseldi 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
416	325, 368, 415, 280, 339	sge0ssrempt 39298	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
417	159, 414	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
418	209, 211, 417	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
419	207, 418	fsumrecl 14312	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
420	336, 342, 415	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
421	202	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
422	75, 67, 144	ltled 10064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ≤ 𝑄)
423	422	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ≤ 𝑄)
424	161, 348, 421, 7, 358, 412, 423, 357, 112, 31	hsphoidmvle2 39475	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
425	336, 342, 424	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
426	325, 327, 363, 420, 425	sge0lempt 39303	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ≤ (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
427	209	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝜑)
428	211	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝑗 ∈ ℕ)
429		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) = 0)
430		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘𝑗) = 0 → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
431	430	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
432	174	mul01d 10114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑄 − 𝑆) · 0) = 0)
433	432	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · 0) = 0)
434	431, 433	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = 0)
435	434	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0))
436	390	addid1d 10115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
437	436	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
438	435, 437	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
439	424	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
440	438, 439	eqbrtrd 4605	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
441	427, 428, 429, 440	syl21anc 1317	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
442		simpl 472	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝜑 ∧ 𝑗 ∈ (1...𝑀)))
443		neqne 2790	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑃‘𝑗) = 0 → (𝑃‘𝑗) ≠ 0)
444	443	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) ≠ 0)
445	405	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
446	209	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝜑)
447	211	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ ℕ)
448		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) ≠ 0)
449	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑋 ∖ 𝑌))
450	201	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ¬ 𝑍 ∈ 𝑌)
451		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘)))
452	161, 219, 449, 450, 7, 112, 359, 451	hoiprodp1 39478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))))
453	452	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))))
454	218	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
455	219	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ∈ Fin)
456	218	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
457		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
458	457	oveqd 6566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑌 = ∅ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
459	458	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
460	250	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):𝑌⟶ℝ)
461		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑌 = ∅ → 𝑌 = ∅)
462	461	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
463	462	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ∅ = 𝑌)
464	463	feq2d 5944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗):∅⟶ℝ ↔ (𝐽‘𝑗):𝑌⟶ℝ))
465	460, 464	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):∅⟶ℝ)
466	271	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):𝑌⟶ℝ)
467	463	feq2d 5944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐾‘𝑗):∅⟶ℝ ↔ (𝐾‘𝑗):𝑌⟶ℝ))
468	466, 467	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):∅⟶ℝ)
469	161, 465, 468	hoidmv0val 39473	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)) = 0)
470	456, 459, 469	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
471	470	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
472		neneq 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑗) ≠ 0 → ¬ (𝑃‘𝑗) = 0)
473	472	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → ¬ (𝑃‘𝑗) = 0)
474	471, 473	pm2.65da 598	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ 𝑌 = ∅)
475	474	neqned 2789	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ≠ ∅)
476	250	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗):𝑌⟶ℝ)
477	271	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗):𝑌⟶ℝ)
478	161, 455, 475, 476, 477	hoidmvn0val 39474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
479	248	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
480	218	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
481	248	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
482	481, 232	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = 𝐹)
483	269	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
484	483, 258	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = 𝐹)
485	482, 484	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = (𝐹(𝐿‘𝑌)𝐹))
486	161, 164, 229	hoidmvval0b 39480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝐹(𝐿‘𝑌)𝐹) = 0)
487	486	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐹(𝐿‘𝑌)𝐹) = 0)
488	480, 485, 487	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
489	488	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
490	472	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ¬ (𝑃‘𝑗) = 0)
491	489, 490	condan 831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
492	491	iftrued 4044	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
493	479, 492	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
494	493	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
495	494	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
496		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
497	496	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
498	495, 497	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
499	269	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
500	491, 253	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
501	499, 500	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
502	501	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
503	502	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
504		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
505	504	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
506	503, 505	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
507	498, 506	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
508	507	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
509	508	prodeq2dv 14492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
510	478, 509	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
511	358	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑆 ∈ ℝ)
512	348	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑊 ∈ Fin)
513	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (𝐷‘𝑗):𝑊⟶ℝ)
514		elun1 3742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
515	514, 7	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
516	515	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
517	357, 511, 512, 513, 516	hsphoival 39469	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)))
518		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)) = ((𝐷‘𝑗)‘𝑘))
519	518	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)) = ((𝐷‘𝑗)‘𝑘))
520	517, 519	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
521	520	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
522	521	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
523	522	prodeq2dv 14492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
524	523	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))))
525	524	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))))
526	454, 510, 525	3eqtrrd 2649	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
527	357, 358, 348, 31, 32	hsphoival 39469	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
528	201	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
529	528	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
530	527, 529	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
531	530	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
532	531	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
533	113	rexrd 9968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ*)
534	533	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ*)
535	33	rexrd 9968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ*)
536	535	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ*)
537		icoltub 38579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷‘𝑗)‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝑆 < ((𝐷‘𝑗)‘𝑍))
538	534, 536, 491, 537	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 < ((𝐷‘𝑗)‘𝑍))
539	358	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ ℝ)
540	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
541	539, 540	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑆 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆))
542	538, 541	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆)
543	542	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆) = 𝑆)
544	543	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = (((𝐶‘𝑗)‘𝑍)[,)𝑆))
545	532, 544	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑆))
546	545	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)))
547		volico 38876	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
548	113, 539, 547	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0)) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
549	548	anabss5 853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
550		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
551	550	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
552		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ ((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
553	552	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
554		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (𝜑 ∧ 𝑗 ∈ ℕ))
555		icogelb 12096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷‘𝑗)‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
556	534, 536, 491, 555	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
557	556	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
558		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆)
559	557, 558	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (((𝐶‘𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆))
560	554, 113	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
561	554, 358	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → 𝑆 ∈ ℝ)
562	560, 561	eqleltd 10060	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (((𝐶‘𝑗)‘𝑍) = 𝑆 ↔ (((𝐶‘𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆)))
563	559, 562	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) = 𝑆)
564		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → ((𝐶‘𝑗)‘𝑍) = 𝑆)
565	564	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → 𝑆 = ((𝐶‘𝑗)‘𝑍))
566	565	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
567	566	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
568	388, 113	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
569	568	subidd 10259	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
570	569	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
571	567, 570	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
572	554, 563, 571	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
573	553, 572	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
574	551, 573	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
575	546, 549, 574	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
576	526, 575	oveq12d 6567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))) = ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))))
577	389, 273	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℂ)
578	358, 113	resubcld 10337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℝ)
579	578	recnd 9947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
580	577, 579	mulcomd 9940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
581	580	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
582	453, 576, 581	3eqtrd 2648	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
583	582	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
584	174	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℂ)
585	579, 584, 577	adddird 9944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
586	585	eqcomd 2616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)))
587	586	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)))
588	579, 584	addcomd 10117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))))
589	153	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℂ)
590	154	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℂ)
591	589, 590, 568	npncand 10295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
592	588, 591	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
593	592	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
594	593	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
595	583, 587, 594	3eqtrd 2648	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
596	446, 447, 448, 595	syl21anc 1317	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
597		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘)))
598	161, 219, 32, 450, 7, 112, 413, 597	hoiprodp1 39478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
599	209, 211, 598	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
600	599	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
601	510	eqcomd 2616	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
602	412	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑄 ∈ ℝ)
603	357, 602, 512, 513, 516	hsphoival 39469	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)))
604		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)) = ((𝐷‘𝑗)‘𝑘))
605	604	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)) = ((𝐷‘𝑗)‘𝑘))
606	603, 605	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
607	606	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
608	607	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
609	608	prodeq2dv 14492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
610	609	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
611	601, 610, 454	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
612	446, 447, 448, 611	syl21anc 1317	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
613	357, 412, 348, 31, 32	hsphoival 39469	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
614	211, 613	syldan 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
615	614	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
616	201	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄))
617	616	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄))
618	211, 33	syldan 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
619	618	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
620		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) = 𝑄)
621	619, 620	eqled 10019	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
622	621	iftrued 4044	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = ((𝐷‘𝑗)‘𝑍))
623	622, 620	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
624	623	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
625	67	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℝ)
626	625	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
627	626	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
628	618	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
629	628	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
630	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 = inf(𝑉, ℝ, < ))
631	446, 39	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑉 ⊆ ℝ)
632	148	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
633		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ (1...𝑀))
634	210, 491	sylanl2 681	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
635	633, 634	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
636		rabid 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↔ (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
637	635, 636	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))})
638		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
639		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
640	639	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
641	640	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 = 𝑗 → (((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍) ↔ ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍)))
642	641	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍)) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
643	637, 638, 642	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
644		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐷‘𝑗)‘𝑍) ∈ V
645	644	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ V)
646	16, 643, 645	elrnmptd 38361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)))
647	646, 14	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑂)
648		elun2 3743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐷‘𝑗)‘𝑍) ∈ 𝑂 → ((𝐷‘𝑗)‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
649	647, 648	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
650	59	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ({(𝐵‘𝑍)} ∪ 𝑂) = 𝑉)
651	649, 650	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑉)
652		lbinfle 10857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ((𝐷‘𝑗)‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ ((𝐷‘𝑗)‘𝑍))
653	631, 632, 651, 652	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → inf(𝑉, ℝ, < ) ≤ ((𝐷‘𝑗)‘𝑍))
654	630, 653	eqbrtrd 4605	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
655	654	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
656		neqne 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ ((𝐷‘𝑗)‘𝑍) = 𝑄 → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
657	656	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
658	627, 629, 655, 657	leneltd 10070	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 < ((𝐷‘𝑗)‘𝑍))
659	627, 629	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → (𝑄 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄))
660	658, 659	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
661	660	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
662	624, 661	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
663	615, 617, 662	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = 𝑄)
664	663	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑄))
665	664	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍))) = (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)))
666	209, 211, 113	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
667	666	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
668	446, 67	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 ∈ ℝ)
669		volico 38876	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)) = if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0))
670	667, 668, 669	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)) = if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0))
671	446, 75	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ ℝ)
672	446, 447, 448, 556	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
673	446, 144	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 < 𝑄)
674	667, 671, 668, 672, 673	lelttrd 10074	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) < 𝑄)
675	674	iftrued 4044	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
676	665, 670, 675	3eqtrd 2648	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍))) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
677	612, 676	oveq12d 6567	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))) = ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))))
678	209, 153	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℂ)
679	388, 666	sseldi 3566	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
680	678, 679	subcld 10271	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
681	306, 680	mulcomd 9940	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
682	681	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
683	600, 677, 682	3eqtrd 2648	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
684	596, 683	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
685	445, 684	eqled 10019	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
686	442, 444, 685	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
687	441, 686	pm2.61dan 828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
688	207, 405, 418, 687	fsumle 14372	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
689	370, 406, 416, 419, 426, 688	leadd12dd 38473	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
690	322	mpteq1d 4666	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))) = (𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
691	690	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
692	211, 415	syldan 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
693	325, 327, 329, 333, 420, 692	sge0splitmpt 39304	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
694	691, 693	eqtrd 2644	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
695	209, 211, 414	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
696	207, 695	sge0fsummpt 39283	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
697	696, 419	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
698		rexadd 11937	. . . . . . . . . . . 12 ⊢ (((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ ∧ (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ) → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
699	416, 697, 698	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +𝑒 (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
700	696	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
701	694, 699, 700	3eqtrrd 2649	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
702	689, 701	breqtrd 4609	. . . . . . . . 9 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
703	407, 280, 200, 411, 702	lemul2ad 10843	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
704	402, 703	eqbrtrd 4605	. . . . . . 7 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
705	196, 279, 281, 315, 704	letrd 10073	. . . . . 6 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
706	180, 705	eqbrtrd 4605	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
707	152, 706	jca 553	. . . 4 ⊢ (𝜑 → (𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
708		oveq1 6556	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (𝑧 − (𝐴‘𝑍)) = (𝑄 − (𝐴‘𝑍)))
709	708	oveq2d 6565	. . . . . 6 ⊢ (𝑧 = 𝑄 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑄 − (𝐴‘𝑍))))
710		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑄 → (𝐻‘𝑧) = (𝐻‘𝑄))
711	710	fveq1d 6105	. . . . . . . . . 10 ⊢ (𝑧 = 𝑄 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑄)‘(𝐷‘𝑗)))
712	711	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑧 = 𝑄 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
713	712	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
714	713	fveq2d 6107	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
715	714	oveq2d 6565	. . . . . 6 ⊢ (𝑧 = 𝑄 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
716	709, 715	breq12d 4596	. . . . 5 ⊢ (𝑧 = 𝑄 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
717	716	elrab 3331	. . . 4 ⊢ (𝑄 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
718	707, 717	sylibr 223	. . 3 ⊢ (𝜑 → 𝑄 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
719	718, 68	syl6eleqr 2699	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
720		breq2 4587	. . 3 ⊢ (𝑢 = 𝑄 → (𝑆 < 𝑢 ↔ 𝑆 < 𝑄))
721	720	rspcev 3282	. 2 ⊢ ((𝑄 ∈ 𝑈 ∧ 𝑆 < 𝑄) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
722	719, 144, 721	syl2anc 691	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   Or wor 4958  ran crn 5039   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑_𝑚 cmap 7744  Fincfn 7841  infcinf 8230  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℤ_≥cuz 11563  ℝ⁺crp 11708   +_𝑒 cxad 11820  [,)cico 12048  [,]cicc 12049  ...cfz 12197  Σcsu 14264  ∏cprod 14474  volcvol 23039  Σ^{^}csumge0 39255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256

This theorem is referenced by:  hoidmvlelem3  39487