dvnprodlem1 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem1 38836

Description: 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnprodlem1.c	⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
dvnprodlem1.j	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
dvnprodlem1.d	⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
dvnprodlem1.t	⊢ (𝜑 → 𝑇 ∈ Fin)
dvnprodlem1.z	⊢ (𝜑 → 𝑍 ∈ 𝑇)
dvnprodlem1.zr	⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
dvnprodlem1.rzt	⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)

**Assertion**
Ref	Expression
dvnprodlem1	⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))

Distinct variable groups: 𝐶,𝑐,𝑘 𝐷,𝑐,𝑡 𝐽,𝑐,𝑘,𝑛,𝑡 𝑅,𝑐,𝑘,𝑛,𝑡 𝑅,𝑠,𝑐,𝑛,𝑡 𝑡,𝑇,𝑠 𝑍,𝑐,𝑘,𝑛,𝑡 𝑍,𝑠 𝜑,𝑐,𝑛,𝑡,𝑘 𝜑,𝑠 Allowed substitution hints: 𝐶(𝑡,𝑛,𝑠) 𝐷(𝑘,𝑛,𝑠) 𝑇(𝑘,𝑛,𝑐) 𝐽(𝑠)

Proof of Theorem dvnprodlem1 Dummy variables 𝑑 𝑒 𝑚 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2611	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
2		0zd 11266	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ∈ ℤ)
3		dvnprodlem1.j	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
4	3	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ ℤ)
5	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℤ)
6		fzssz 12214	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℤ
7	6	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℤ)
8		dvnprodlem1.c	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
9	8	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})))
10		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})))
11		rabeq 3166	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
12	10, 11	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
13		sumeq1 14267	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
14	13	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛))
15	14	rabbidv 3164	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
16	12, 15	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
17	16	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
18	17	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 = (𝑅 ∪ {𝑍})) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
19		dvnprodlem1.rzt	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
20		dvnprodlem1.t	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑇 ∈ Fin)
21		ssexg 4732	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ∪ {𝑍}) ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ V)
22	19, 20, 21	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
23		elpwg 4116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
25	19, 24	mpbird 246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
26		nn0ex 11175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ₀ ∈ V
27	26	mptex 6390	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V)
29	9, 18, 25, 28	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
30		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
31	30	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
32		rabeq 3166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
34		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
35	34	rabbidv 3164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
36	33, 35	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 = 𝐽) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
38		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∈ V
39	38	rabex 4740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V)
41	29, 37, 3, 40	fvmptd 6197	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
42		ssrab2 3650	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍}))
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
44	41, 43	eqsstrd 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
45	44	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
46		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
47	45, 46	sseldd 3569	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
48		elmapi 7765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
50		dvnprodlem1.z	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
51		snidg 4153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍})
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
53		elun2 3743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍}))
55	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
56	49, 55	ffvelrnd 6268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽))
57	7, 56	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℤ)
58	5, 57	zsubcld 11363	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
59	2, 5, 58	3jca 1235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ))
60		elfzle2 12216	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽)
61	56, 60	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ≤ 𝐽)
62	5	zred 11358	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℝ)
63	57	zred 11358	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℝ)
64	62, 63	subge0d 10496	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽))
65	61, 64	mpbird 246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍)))
66		elfzle1 12215	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍))
67	56, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝑐‘𝑍))
68	62, 63	subge02d 10498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))
69	67, 68	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)
70	59, 65, 69	jca32 556	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
71		elfz2 12204	. . . . . . . . . . 11 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
72	70, 71	sylibr 223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
73		elmapfn 7766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → 𝑐 Fn (𝑅 ∪ {𝑍}))
74	47, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
75		ssun1 3738	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 ⊆ (𝑅 ∪ {𝑍})
76	75	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
77		fnssres 5918	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑅 ⊆ (𝑅 ∪ {𝑍})) → (𝑐 ↾ 𝑅) Fn 𝑅)
78	74, 76, 77	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) Fn 𝑅)
79		nfv 1830	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡𝜑
80		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡𝑐
81		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑡𝒫 𝑇
82		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑡ℕ₀
83		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑡𝑠
84	83	nfsum1 14268	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡)
85		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑡𝑛
86	84, 85	nfeq 2762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛
87		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑡((0...𝑛) ↑_𝑚 𝑠)
88	86, 87	nfrab 3100	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑡{𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}
89	82, 88	nfmpt 4674	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
90	81, 89	nfmpt 4674	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑡(𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
91	8, 90	nfcxfr 2749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡𝐶
92		nfcv 2751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡(𝑅 ∪ {𝑍})
93	91, 92	nffv 6110	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡(𝐶‘(𝑅 ∪ {𝑍}))
94		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡𝐽
95	93, 94	nffv 6110	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
96	80, 95	nfel 2763	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
97	79, 96	nfan 1816	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
98		fvres 6117	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
99	98	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
100	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ∈ ℤ)
101	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
102	6	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℤ)
103	49	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
104	76	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
105	103, 104	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽))
106	102, 105	sseldd 3569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℤ)
107	100, 101, 106	3jca 1235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ))
108		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐‘𝑡) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑡))
109	105, 108	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ≤ (𝑐‘𝑡))
110	19	unssad 3752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑅 ⊆ 𝑇)
111		ssfi 8065	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin)
112	20, 110, 111	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ∈ Fin)
113	112	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑅 ∈ Fin)
114		zssre 11261	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℤ ⊆ ℝ
115	6, 114	sstri 3577	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0...𝐽) ⊆ ℝ
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (0...𝐽) ⊆ ℝ)
117	49	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
118	76	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ (𝑅 ∪ {𝑍}))
119	117, 118	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ (0...𝐽))
120	116, 119	sseldd 3569	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ)
121	120	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ)
122		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑐‘𝑟) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑟))
123	119, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟))
124	123	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟))
125		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑟 = 𝑡 → (𝑐‘𝑟) = (𝑐‘𝑡))
126		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅)
127	113, 121, 124, 125, 126	fsumge1 14370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
128	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
129	120	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℂ)
130	128, 129	fsumcl 14311	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) ∈ ℂ)
131	63	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ)
132	130, 131	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
133		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑟(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
134		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑟(𝑐‘𝑍)
135	50	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
136		dvnprodlem1.zr	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
137	136	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
138		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑟 = 𝑍 → (𝑐‘𝑟) = (𝑐‘𝑍))
139	133, 134, 128, 135, 137, 129, 138, 131	fsumsplitsn 38637	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)))
140	139	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟))
141	125	cbvsumv 14274	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)
142	141	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
143	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
144	46, 143	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
145		rabid 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
146	144, 145	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
147	146	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)
148	142, 147	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = 𝐽)
149	140, 148	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = 𝐽)
150	149	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = (𝐽 − (𝑐‘𝑍)))
151	132, 150	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
152	151	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
153	127, 152	breqtrd 4609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))
154	107, 109, 153	jca32 556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))))
155		elfz2 12204	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))) ↔ ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))))
156	154, 155	sylibr 223	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
157	99, 156	eqeltrd 2688	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
158	157	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
159	97, 158	ralrimi 2940	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
160	78, 159	jca 553	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
161		ffnfv 6295	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))) ↔ ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
162	160, 161	sylibr 223	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))))
163		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ (0...(𝐽 − (𝑐‘𝑍))) ∈ V
164	163	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...(𝐽 − (𝑐‘𝑍))) ∈ V)
165	20, 110	ssexd 4733	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ V)
166	165	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ V)
167	164, 166	elmapd 7758	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ↔ (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍)))))
168	162, 167	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅))
169	98	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)))
170	97, 169	ralrimi 2940	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
171	170	sumeq2d 14280	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
172	125	cbvsumv 14274	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)
173	172	eqcomi 2619	. . . . . . . . . . . . . . 15 ⊢ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)
174	173	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
175	151	idi 2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
176	171, 174, 175	3eqtrd 2648	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))
177	168, 176	jca 553	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
178		eqidd 2611	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → 𝑅 = 𝑅)
179		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → 𝑒 = (𝑐 ↾ 𝑅))
180	179	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → (𝑒‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
181	178, 180	sumeq12rdv 14285	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡))
182	181	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)) ↔ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
183	182	elrab 3331	. . . . . . . . . . . 12 ⊢ ((𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ↔ ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
184	177, 183	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
185		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑅))
186		rabeq 3166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
187	185, 186	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
188		sumeq1 14267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
189	188	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛))
190	189	rabbidv 3164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
191	187, 190	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
192	191	mpteq2dv 4673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑅 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
193	192	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 = 𝑅) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
194		elpwg 4116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
195	165, 194	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
196	110, 195	mpbird 246	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇)
197	26	mptex 6390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V
198	197	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V)
199	9, 193, 196, 198	fvmptd 6197	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
200	199	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
201		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
202	201	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑚) ↑_𝑚 𝑅))
203		rabeq 3166	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑚) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
204	202, 203	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
205		eqeq2 2621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚))
206	205	rabbidv 3164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
207	204, 206	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
208	207	cbvmptv 4678	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
209	208	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}))
210	200, 209	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}))
211		fveq1 6102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
212	211	sumeq2ad 38632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑒‘𝑡))
213	212	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚))
214	213	cbvrabv 3172	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}
215	214	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
216		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (0...𝑚) = (0...(𝐽 − (𝑐‘𝑍))))
217	216	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → ((0...𝑚) ↑_𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅))
218		rabeq 3166	. . . . . . . . . . . . . . . 16 ⊢ (((0...𝑚) ↑_𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) → {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
219	217, 218	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
220		eqeq2 2621	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))))
221	220	rabbidv 3164	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
222	215, 219, 221	3eqtrd 2648	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
223	222	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑚 = (𝐽 − (𝑐‘𝑍))) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
224	58, 65	jca 553	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍))))
225		elnn0z 11267	. . . . . . . . . . . . . 14 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ ℕ₀ ↔ ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍))))
226	224, 225	sylibr 223	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
227		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∈ V
228	227	rabex 4740	. . . . . . . . . . . . . 14 ⊢ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V
229	228	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V)
230	210, 223, 226, 229	fvmptd 6197	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))) = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
231	230	eqcomd 2616	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
232	184, 231	eleqtrd 2690	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
233	72, 232	jca 553	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))
234	1, 233	jca 553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))))
235	232	elfvexd 6132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ V)
236		vex 3176	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
237	236	resex 5363	. . . . . . . . . 10 ⊢ (𝑐 ↾ 𝑅) ∈ V
238	237	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ V)
239		opeq12 4342	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ⟨𝑘, 𝑑⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
240	239	eqeq2d 2620	. . . . . . . . . . 11 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ↔ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
241		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)))
242	241	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)))
243		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 𝑑 = (𝑐 ↾ 𝑅))
244		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
245	244	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
246	243, 245	eleq12d 2682	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑑 ∈ ((𝐶‘𝑅)‘𝑘) ↔ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))
247	242, 246	anbi12d 743	. . . . . . . . . . 11 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))))
248	240, 247	anbi12d 743	. . . . . . . . . 10 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))) ↔ (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))))
249	248	spc2egv 3268	. . . . . . . . 9 ⊢ (((𝐽 − (𝑐‘𝑍)) ∈ V ∧ (𝑐 ↾ 𝑅) ∈ V) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))))
250	235, 238, 249	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))))
251	234, 250	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))
252		eliunxp 5181	. . . . . . 7 ⊢ (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))
253	251, 252	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
254		dvnprodlem1.d	. . . . . 6 ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
255	253, 254	fmptd 6292	. . . . 5 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
256	95	nfcri 2745	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
257	96, 256	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
258	79, 257	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
259		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝐷‘𝑐) = (𝐷‘𝑒)
260	258, 259	nfan 1816	. . . . . . . . 9 ⊢ Ⅎ𝑡((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒))
261	99	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
262	261	adantlrr 753	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
263	262	adantlr 747	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
264	254	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
265		opex 4859	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V
266	265	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V)
267	264, 266	fvmpt2d 6202	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
268	267	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
269	268	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡))
270		ovex 6577	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 − (𝑐‘𝑍)) ∈ V
271	270, 237	op2nd 7068	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝑐 ↾ 𝑅)
272	271	fveq1i 6104	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)
273	272	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
274	269, 273	eqtr2d 2645	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
275	274	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
276	275	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
277		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = (𝐷‘𝑒))
278	254	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
279		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑍) = (𝑒‘𝑍))
280	279	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝑒‘𝑍)))
281		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝑐 ↾ 𝑅) = (𝑒 ↾ 𝑅))
282	280, 281	opeq12d 4348	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑒 → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
283	282	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑐 = 𝑒) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
284		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
285		opex 4859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩ ∈ V
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩ ∈ V)
287	278, 283, 284, 286	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑒) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
288	287	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑒) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
289	277, 288	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
290	289	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
291	290	adantlrl 752	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
292	291	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
293		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 − (𝑒‘𝑍)) ∈ V
294		vex 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑒 ∈ V
295	294	resex 5363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ↾ 𝑅) ∈ V
296	293, 295	op2nd 7068	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝑒 ↾ 𝑅)
297	296	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝑒 ↾ 𝑅))
298	292, 297	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘(𝐷‘𝑐)) = (𝑒 ↾ 𝑅))
299	298	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = ((𝑒 ↾ 𝑅)‘𝑡))
300		fvres 6117	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑅 → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡))
301	300	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡))
302	299, 301	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = (𝑒‘𝑡))
303	263, 276, 302	3eqtrd 2648	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
304	303	adantlr 747	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
305		simpl 472	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
306		elunnel1 3716	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 ∈ {𝑍})
307		elsni 4142	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ {𝑍} → 𝑡 = 𝑍)
308	306, 307	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
309	308	adantll 746	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
310		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍)
311	310	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑐‘𝑍))
312	3	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ ℂ)
313	312	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
314	313, 131	nncand 10276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍))
315	314	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
316	315	adantrr 749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
317	316	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
318	267	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
319	270, 237	op1st 7067	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝐽 − (𝑐‘𝑍))
320	319	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝐽 − (𝑐‘𝑍)))
321	318, 320	eqtr2d 2645	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) = (1^st ‘(𝐷‘𝑐)))
322	321	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
323	322	adantrr 749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
324	323	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
325		fveq2 6103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷‘𝑐) = (𝐷‘𝑒) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘(𝐷‘𝑒)))
326	325	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘(𝐷‘𝑒)))
327	287	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘(𝐷‘𝑒)) = (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
328	327	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑒)) = (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
329	293, 295	op1st 7067	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝐽 − (𝑒‘𝑍))
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝐽 − (𝑒‘𝑍)))
331	326, 328, 330	3eqtrd 2648	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑐)) = (𝐽 − (𝑒‘𝑍)))
332	331	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝐽 − (𝐽 − (𝑒‘𝑍))))
333	312	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
334		zsscn 11262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℤ ⊆ ℂ
335	6, 334	sstri 3577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝐽) ⊆ ℂ
336	335	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
337		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑒 → (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↔ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
338	337	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ↔ (𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))))
339		feq1 5939	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽) ↔ 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)))
340	338, 339	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑒 → (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) ↔ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))))
341	340, 49	chvarv 2251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))
342	54	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
343	341, 342	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ (0...𝐽))
344	336, 343	sseldd 3569	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ ℂ)
345	333, 344	nncand 10276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍))
346	345	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍))
347		eqidd 2611	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑒‘𝑍) = (𝑒‘𝑍))
348	332, 346, 347	3eqtrd 2648	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝑒‘𝑍))
349	348	adantlrl 752	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝑒‘𝑍))
350	317, 324, 349	3eqtrd 2648	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝑒‘𝑍))
351	350	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑍) = (𝑒‘𝑍))
352		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → (𝑒‘𝑡) = (𝑒‘𝑍))
353	352	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (𝑒‘𝑍) = (𝑒‘𝑡))
354	353	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑒‘𝑍) = (𝑒‘𝑡))
355	311, 351, 354	3eqtrd 2648	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡))
356	355	adantlr 747	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡))
357	305, 309, 356	syl2anc 691	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
358	304, 357	pm2.61dan 828	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) = (𝑒‘𝑡))
359	358	ex 449	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → (𝑐‘𝑡) = (𝑒‘𝑡)))
360	260, 359	ralrimi 2940	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))
361	74	adantrr 749	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑐 Fn (𝑅 ∪ {𝑍}))
362	361	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
363		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽) → 𝑒 Fn (𝑅 ∪ {𝑍}))
364	341, 363	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
365	364	adantrl 748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑒 Fn (𝑅 ∪ {𝑍}))
366	365	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
367		eqfnfv 6219	. . . . . . . . 9 ⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑒 Fn (𝑅 ∪ {𝑍})) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)))
368	362, 366, 367	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)))
369	360, 368	mpbird 246	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 = 𝑒)
370	369	ex 449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒))
371	370	ralrimivva 2954	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒))
372	255, 371	jca 553	. . . 4 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)))
373		dff13 6416	. . . 4 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)))
374	372, 373	sylibr 223	. . 3 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
375		eliun 4460	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
376	375	biimpi 205	. . . . . . . . . 10 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
377	376	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
378		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑
379		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑝
380		nfiu1 4486	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
381	379, 380	nfel 2763	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
382	378, 381	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
383		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}
384		nfv 1830	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑘 ∈ (0...𝐽)
385		nfcv 2751	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡𝑝
386		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡{𝑘}
387		nfcv 2751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡𝑅
388	91, 387	nffv 6110	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡(𝐶‘𝑅)
389		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡𝑘
390	388, 389	nffv 6110	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡((𝐶‘𝑅)‘𝑘)
391	386, 390	nfxp 5066	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡({𝑘} × ((𝐶‘𝑅)‘𝑘))
392	385, 391	nfel 2763	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))
393	79, 384, 392	nf3an 1819	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
394		0zd 11266	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ∈ ℤ)
395	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
396	395	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
397		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
398	397	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
399		fzssz 12214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...𝑘) ⊆ ℤ
400	399	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℤ)
401		simp1 1054	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
402		simp2 1055	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
403		xp2nd 7090	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
404	403	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
405	199	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
406		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
407	406	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑘) ↑_𝑚 𝑅))
408		rabeq 3166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑘) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
409	407, 408	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
410		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘))
411	410	rabbidv 3164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
412	409, 411	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
413	412	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
414		elfznn0 12302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ₀)
415	414	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ₀)
416		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((0...𝑘) ↑_𝑚 𝑅) ∈ V
417	416	rabex 4740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V
418	417	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V)
419	405, 413, 415, 418	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
420	419	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
421	404, 420	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
422		elrabi 3328	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
423	422	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ (2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
424	401, 402, 421, 423	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
425		elmapi 7765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
426	424, 425	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
427	426	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
428	427	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘))
429	400, 428	sseldd 3569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
430	398, 429	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
431		simpl 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
432	308	adantll 746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
433		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍)
434	136	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑍 ∈ 𝑅)
435	433, 434	eqneltrd 2707	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅)
436	435	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
437	436	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
438	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ)
439	438	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ)
440		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ {𝑘})
441		elsni 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((1^st ‘𝑝) ∈ {𝑘} → (1^st ‘𝑝) = 𝑘)
442	440, 441	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) = 𝑘)
443	442	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
444	6	sseli 3564	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
445	444	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℤ)
446	443, 445	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
447	446	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
448	447	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (1^st ‘𝑝) ∈ ℤ)
449	439, 448	zsubcld 11363	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1^st ‘𝑝)) ∈ ℤ)
450	437, 449	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
451	450	adantlr 747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
452	431, 432, 451	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
453	430, 452	pm2.61dan 828	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
454	394, 396, 453	3jca 1235	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ))
455	426	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘))
456		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → 0 ≤ ((2^nd ‘𝑝)‘𝑡))
457	455, 456	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ ((2^nd ‘𝑝)‘𝑡))
458	397	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝑅 → ((2^nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
459	458	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
460	457, 459	breqtrd 4609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
461	460	adantlr 747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
462		simpll 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))))
463		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽)
464		elfzel2 12211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
465	464	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
466	115	sseli 3564	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
467	465, 466	subge0d 10496	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽))
468	463, 467	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘))
469	468	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘))
470	469	3ad2antl2 1217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘))
471	401, 435	sylan 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅)
472	471	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
473	443	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
474	473	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
475	474	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
476	472, 475	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
477	470, 476	breqtrd 4609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
478	462, 432, 477	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
479	461, 478	pm2.61dan 828	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
480		simpl2 1058	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑘 ∈ (0...𝐽))
481	399	sseli 3564	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
482	481	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ∈ ℝ)
483	482	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ∈ ℝ)
484	466	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
485	465	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
486		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝑘)
487	486	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝑘)
488	463	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽)
489	483, 484, 485, 487, 488	letrd 10073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
490	455, 480, 489	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
491	490	adantlr 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
492	398, 491	eqbrtrd 4605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
493	476	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − 𝑘))
494	415	nn0ge0d 11231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
495	465	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
496	466	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
497	495, 496	subge02d 10498	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽))
498	494, 497	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ≤ 𝐽)
499	498	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽)
500	499	3adantl3 1212	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽)
501	493, 500	eqbrtrd 4605	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
502	462, 432, 501	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
503	492, 502	pm2.61dan 828	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
504	454, 479, 503	jca32 556	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ) ∧ (0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)))
505		elfz2 12204	. . . . . . . . . . . . . . . . 17 ⊢ (if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ) ∧ (0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)))
506	504, 505	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ (0...𝐽))
507		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
508	393, 506, 507	fmptdf 6294	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽))
509		ovex 6577	. . . . . . . . . . . . . . . . 17 ⊢ (0...𝐽) ∈ V
510	509	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ∈ V)
511	401, 22	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑅 ∪ {𝑍}) ∈ V)
512	510, 511	elmapd 7758	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ↔ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽)))
513	508, 512	mpbird 246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
514		eqidd 2611	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
515		eleq1 2676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑡 → (𝑟 ∈ 𝑅 ↔ 𝑡 ∈ 𝑅))
516		fveq2 6103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑡 → ((2^nd ‘𝑝)‘𝑟) = ((2^nd ‘𝑝)‘𝑡))
517	515, 516	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑡 → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
518	517	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑟 = 𝑡) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
519		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
520	514, 518, 519, 453	fvmptd 6197	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
521	520	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
522	393, 521	ralrimi 2940	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
523	522	sumeq2d 14280	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
524		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))
525	401, 112	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
526	401, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇)
527	401, 136	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅)
528	397	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
529	455, 481	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
530	529	zcnd 11359	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℂ)
531	528, 530	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℂ)
532		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑅 ↔ 𝑍 ∈ 𝑅))
533		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑍 → ((2^nd ‘𝑝)‘𝑡) = ((2^nd ‘𝑝)‘𝑍))
534	532, 533	ifbieq1d 4059	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))))
535	136	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅)
536	535	iffalsed 4047	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
537	536	3adant2 1073	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
538	4	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℤ)
539	538, 447	zsubcld 11363	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ ℤ)
540	539	zcnd 11359	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ ℂ)
541	537, 540	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) ∈ ℂ)
542	393, 524, 525, 526, 527, 531, 534, 541	fsumsplitsn 38637	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))))
543	397	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡)))
544	393, 543	ralrimi 2940	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
545	544	sumeq2d 14280	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡))
546		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (2^nd ‘𝑝) → 𝑅 = 𝑅)
547		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 = (2^nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → 𝑐 = (2^nd ‘𝑝))
548	547	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 = (2^nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((2^nd ‘𝑝)‘𝑡))
549	546, 548	sumeq12rdv 14285	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (2^nd ‘𝑝) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡))
550	549	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (2^nd ‘𝑝) → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘 ↔ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
551	550	elrab 3331	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ↔ ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
552	421, 551	sylib 207	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
553	552	simprd 478	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘)
554	545, 553	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = 𝑘)
555	527	iffalsed 4047	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
556	555, 474	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − 𝑘))
557	554, 556	oveq12d 6567	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))) = (𝑘 + (𝐽 − 𝑘)))
558	335	sseli 3564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℂ)
559	558	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℂ)
560	401, 312	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℂ)
561	559, 560	pncan3d 10274	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 + (𝐽 − 𝑘)) = 𝐽)
562	557, 561	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))) = 𝐽)
563	523, 542, 562	3eqtrd 2648	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽)
564	513, 563	jca 553	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
565		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑟 → (𝑡 ∈ 𝑅 ↔ 𝑟 ∈ 𝑅))
566		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑟 → ((2^nd ‘𝑝)‘𝑡) = ((2^nd ‘𝑝)‘𝑟))
567	565, 566	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑟 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))
568	567	cbvmptv 4678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))
569	568	eqeq2i 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↔ 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
570	569	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
571		fveq1 6102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) → (𝑐‘𝑡) = ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
572	571	sumeq2ad 38632	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
573	570, 572	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
574	573	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
575	574	elrab 3331	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
576	564, 575	sylibr 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
577	576	3exp 1256	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})))
578	577	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})))
579	382, 383, 578	rexlimd 3008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}))
580	377, 579	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
581	41	eqcomd 2616	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
582	581	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
583	580, 582	eleqtrd 2690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
584	254	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
585		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
586	568	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
587	585, 586	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
588		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → 𝑟 = 𝑍)
589	136	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑍 ∈ 𝑅)
590	588, 589	eqneltrd 2707	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑟 ∈ 𝑅)
591	590	iffalsed 4047	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
592	591	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
593	54	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
594		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝐽 − (1^st ‘𝑝)) ∈ V
595	594	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (1^st ‘𝑝)) ∈ V)
596	587, 592, 593, 595	fvmptd 6197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐‘𝑍) = (𝐽 − (1^st ‘𝑝)))
597	596	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1^st ‘𝑝))))
598	597	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1^st ‘𝑝))))
599	312	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝐽 ∈ ℂ)
600		nfv 1830	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(1^st ‘𝑝) ∈ (0...𝐽)
601		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
602		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → (1^st ‘𝑝) = 𝑘)
603		simpl 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → 𝑘 ∈ (0...𝐽))
604	602, 603	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → (1^st ‘𝑝) ∈ (0...𝐽))
605	601, 443, 604	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
606	605	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
607	606	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽))))
608	381, 600, 607	rexlimd 3008	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
609	376, 608	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽))
610	6	sseli 3564	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℤ)
611	609, 610	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ ℤ)
612	611	zcnd 11359	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ ℂ)
613	612	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (1^st ‘𝑝) ∈ ℂ)
614	599, 613	nncand 10276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝐽 − (1^st ‘𝑝))) = (1^st ‘𝑝))
615	598, 614	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (1^st ‘𝑝))
616		reseq1 5311	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅))
617	616	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅))
618	75	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
619	618	resmptd 5371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅) = (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
620		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝)
621	397	mpteq2ia 4668	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡))
622	621	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡)))
623	426	feqmptd 6159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡)))
624	622, 623	eqtr4d 2647	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
625	624	3exp 1256	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))))
626	625	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))))
627	382, 620, 626	rexlimd 3008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝)))
628	377, 627	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
629	628	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
630	617, 619, 629	3eqtrd 2648	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐 ↾ 𝑅) = (2^nd ‘𝑝))
631	615, 630	opeq12d 4348	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
632		opex 4859	. . . . . . . . . 10 ⊢ ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∈ V
633	632	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∈ V)
634	584, 631, 583, 633	fvmptd 6197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
635		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝
636		1st2nd2 7096	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
637	636	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)
638	637	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝))
639	638	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)))
640	382, 635, 639	rexlimd 3008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝))
641	377, 640	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)
642	634, 641	eqtr2d 2645	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))))
643		fveq2 6103	. . . . . . . . 9 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝐷‘𝑐) = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))))
644	643	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝑝 = (𝐷‘𝑐) ↔ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))))
645	644	rspcev 3282	. . . . . . 7 ⊢ (((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
646	583, 642, 645	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
647	646	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
648	255, 647	jca 553	. . . 4 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)))
649		dffo3 6282	. . . 4 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)))
650	648, 649	sylibr 223	. . 3 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
651	374, 650	jca 553	. 2 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))))
652		df-f1o 5811	. 2 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))))
653	651, 652	sylibr 223	1 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ifcif 4036 𝒫 cpw 4108 {csn 4125 ⟨cop 4131 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 Σcsu 14264

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-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 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-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 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-sum 14265

This theorem is referenced by: dvnprodlem2 38837

W3C validator