Theorem caucvgprlemladdfu 6648

Description: Lemma for caucvgpr 6653. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (φ → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (φ → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd	⊢ (φ → ∀𝑗 ∈ N A <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
caucvgprlemladd.s	⊢ (φ → 𝑆 ∈ Q)

Assertion

Ref	Expression
caucvgprlemladdfu	⊢ (φ → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) ⊆ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})

Distinct variable groups:   A,𝑗   𝑗,𝐹,u,𝑙   𝑛,𝐹,𝑘   𝑘,𝐿,𝑗   𝑆,𝑙,u,𝑗   𝑗,𝑘

Allowed substitution hints:   φ(u,𝑗,𝑘,𝑛,𝑙)   A(u,𝑘,𝑛,𝑙)   𝑆(𝑘,𝑛)   𝐿(u,𝑛,𝑙)

Proof of Theorem caucvgprlemladdfu

Dummy variables 𝑚 𝑟 𝑠 𝑡 v w z f g ℎ x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . . . . 7 ⊢ (φ → 𝐹:N⟶Q)
2		caucvgpr.cau	. . . . . . 7 ⊢ (φ → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3		caucvgpr.bnd	. . . . . . 7 ⊢ (φ → ∀𝑗 ∈ N A <Q (𝐹‘𝑗))
4		caucvgpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
5	1, 2, 3, 4	caucvgprlemcl 6647	. . . . . 6 ⊢ (φ → 𝐿 ∈ P)
6		caucvgprlemladd.s	. . . . . . 7 ⊢ (φ → 𝑆 ∈ Q)
7		nqprlu 6530	. . . . . . 7 ⊢ (𝑆 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩ ∈ P)
8	6, 7	syl 14	. . . . . 6 ⊢ (φ → ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩ ∈ P)
9		df-iplp 6451	. . . . . . 7 ⊢ +P = (x ∈ P, y ∈ P ↦ ⟨{f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (1st ‘x) ∧ ℎ ∈ (1st ‘y) ∧ f = (g +Q ℎ))}, {f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (2nd ‘x) ∧ ℎ ∈ (2nd ‘y) ∧ f = (g +Q ℎ))}⟩)
10		addclnq 6359	. . . . . . 7 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g +Q ℎ) ∈ Q)
11	9, 10	genpelvu 6496	. . . . . 6 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) ↔ ∃𝑠 ∈ (2nd ‘𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)𝑟 = (𝑠 +Q 𝑡)))
12	5, 8, 11	syl2anc 391	. . . . 5 ⊢ (φ → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) ↔ ∃𝑠 ∈ (2nd ‘𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)𝑟 = (𝑠 +Q 𝑡)))
13	12	biimpa 280	. . . 4 ⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → ∃𝑠 ∈ (2nd ‘𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)𝑟 = (𝑠 +Q 𝑡))
14		breq2 3759	. . . . . . . . . . . . . . . 16 ⊢ (u = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
15	14	rexbidv 2321	. . . . . . . . . . . . . . 15 ⊢ (u = 𝑠 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
16	4	fveq2i 5124	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
17		nqex 6347	. . . . . . . . . . . . . . . . . 18 ⊢ Q ∈ V
18	17	rabex 3892	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
19	17	rabex 3892	. . . . . . . . . . . . . . . . 17 ⊢ {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} ∈ V
20	18, 19	op2nd 5716	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
21	16, 20	eqtri 2057	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐿) = {u ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
22	15, 21	elrab2 2694	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
23	22	biimpi 113	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐿) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
24	23	adantr 261	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
25	24	adantl 262	. . . . . . . . . . 11 ⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
26	25	adantr 261	. . . . . . . . . 10 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
27	26	simpld 105	. . . . . . . . 9 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 ∈ Q)
28		vex 2554	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
29		breq2 3759	. . . . . . . . . . . . . 14 ⊢ (u = 𝑡 → (𝑆 <Q u ↔ 𝑆 <Q 𝑡))
30		ltnqex 6531	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 6532	. . . . . . . . . . . . . . 15 ⊢ {u ∣ 𝑆 <Q u} ∈ V
32	30, 31	op2nd 5716	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩) = {u ∣ 𝑆 <Q u}
33	28, 29, 32	elab2 2684	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩) ↔ 𝑆 <Q 𝑡)
34		ltrelnq 6349	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
35	34	brel 4335	. . . . . . . . . . . . 13 ⊢ (𝑆 <Q 𝑡 → (𝑆 ∈ Q ∧ 𝑡 ∈ Q))
36	33, 35	sylbi 114	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩) → (𝑆 ∈ Q ∧ 𝑡 ∈ Q))
37	36	simprd 107	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩) → 𝑡 ∈ Q)
38	37	ad2antll 460	. . . . . . . . . 10 ⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → 𝑡 ∈ Q)
39	38	adantr 261	. . . . . . . . 9 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 ∈ Q)
40		addclnq 6359	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q) → (𝑠 +Q 𝑡) ∈ Q)
41	27, 39, 40	syl2anc 391	. . . . . . . 8 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈ Q)
42		eleq1 2097	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
43	42	adantl 262	. . . . . . . 8 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
44	41, 43	mpbird 156	. . . . . . 7 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ Q)
45	26	simprd 107	. . . . . . . . . 10 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
46		fveq2 5121	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
47		opeq1 3540	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
48	47	eceq1d 6078	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
49	48	fveq2d 5125	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
50	46, 49	oveq12d 5473	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
51	50	breq1d 3765	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠))
52	51	cbvrexv 2528	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ ∃𝑚 ∈ N ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
53	45, 52	sylib 127	. . . . . . . . 9 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚 ∈ N ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
54	33	biimpi 113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩) → 𝑆 <Q 𝑡)
55	54	ad2antll 460	. . . . . . . . . . . . . . . 16 ⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → 𝑆 <Q 𝑡)
56	55	adantr 261	. . . . . . . . . . . . . . 15 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑆 <Q 𝑡)
57	56	ad2antrr 457	. . . . . . . . . . . . . 14 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑆 <Q 𝑡)
58	6	ad5antr 465	. . . . . . . . . . . . . . 15 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑆 ∈ Q)
59	39	ad2antrr 457	. . . . . . . . . . . . . . 15 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑡 ∈ Q)
60	1	ad5antr 465	. . . . . . . . . . . . . . . . 17 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝐹:N⟶Q)
61		simplr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑚 ∈ N)
62	60, 61	ffvelrnd 5246	. . . . . . . . . . . . . . . 16 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (𝐹‘𝑚) ∈ Q)
63		nnnq 6405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ N → [⟨𝑚, 1𝑜⟩] ~Q ∈ Q)
64		recclnq 6376	. . . . . . . . . . . . . . . . 17 ⊢ ([⟨𝑚, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q)
65	61, 63, 64	3syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q)
66		addclnq 6359	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
67	62, 65, 66	syl2anc 391	. . . . . . . . . . . . . . 15 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
68		ltanqg 6384	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Q ∧ 𝑡 ∈ Q ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡)))
69	58, 59, 67, 68	syl3anc 1134	. . . . . . . . . . . . . 14 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡)))
70	57, 69	mpbid 135	. . . . . . . . . . . . 13 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡))
71		simpr 103	. . . . . . . . . . . . . 14 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
72		ltanqg 6384	. . . . . . . . . . . . . . . 16 ⊢ ((z ∈ Q ∧ w ∈ Q ∧ v ∈ Q) → (z <Q w ↔ (v +Q z) <Q (v +Q w)))
73	72	adantl 262	. . . . . . . . . . . . . . 15 ⊢ (((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) ∧ (z ∈ Q ∧ w ∈ Q ∧ v ∈ Q)) → (z <Q w ↔ (v +Q z) <Q (v +Q w)))
74	27	ad2antrr 457	. . . . . . . . . . . . . . 15 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑠 ∈ Q)
75		addcomnqg 6365	. . . . . . . . . . . . . . . 16 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z +Q w) = (w +Q z))
76	75	adantl 262	. . . . . . . . . . . . . . 15 ⊢ (((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) ∧ (z ∈ Q ∧ w ∈ Q)) → (z +Q w) = (w +Q z))
77	73, 67, 74, 59, 76	caovord2d 5612	. . . . . . . . . . . . . 14 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡)))
78	71, 77	mpbid 135	. . . . . . . . . . . . 13 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡))
79		ltsonq 6382	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
80	79, 34	sotri 4663	. . . . . . . . . . . . 13 ⊢ (((((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) ∧ (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡)) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (𝑠 +Q 𝑡))
81	70, 78, 80	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (𝑠 +Q 𝑡))
82		simpllr 486	. . . . . . . . . . . 12 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
83	81, 82	breqtrrd 3781	. . . . . . . . . . 11 ⊢ ((((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
84	83	ex 108	. . . . . . . . . 10 ⊢ (((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
85	84	reximdva 2415	. . . . . . . . 9 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑚 ∈ N ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 → ∃𝑚 ∈ N (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
86	53, 85	mpd 13	. . . . . . . 8 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚 ∈ N (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
87	50	oveq1d 5470	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) = (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆))
88	87	breq1d 3765	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → ((((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
89	88	cbvrexv 2528	. . . . . . . 8 ⊢ (∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ ∃𝑚 ∈ N (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
90	86, 89	sylibr 137	. . . . . . 7 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
91		breq2 3759	. . . . . . . . 9 ⊢ (u = 𝑟 → ((((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u ↔ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
92	91	rexbidv 2321	. . . . . . . 8 ⊢ (u = 𝑟 → (∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u ↔ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
93	92	elrab 2692	. . . . . . 7 ⊢ (𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u} ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
94	44, 90, 93	sylanbrc 394	. . . . . 6 ⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})
95	94	ex 108	. . . . 5 ⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) ∧ (𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
96	95	rexlimdvva 2434	. . . 4 ⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → (∃𝑠 ∈ (2nd ‘𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
97	13, 96	mpd 13	. . 3 ⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩))) → 𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})
98	97	ex 108	. 2 ⊢ (φ → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) → 𝑟 ∈ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
99	98	ssrdv 2945	1 ⊢ (φ → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}⟩)) ⊆ {u ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  {crab 2304   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  2^nd c2nd 5708  1_𝑜c1o 5933  [cec 6040  Ncnpi 6256   <_N clti 6259   ~_Q ceq 6263  Qcnq 6264   +_Q cplq 6266  *_Qcrq 6268   <_Q cltq 6269  Pcnp 6275   +_P cpp 6277

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-iplp 6451

This theorem is referenced by:  caucvgprlemladdrl  6649