Theorem caucvgprlemladdfu 7678

Description: Lemma for caucvgpr 7683. 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 7677	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		caucvgprlemladd.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Q)
7		nqprlu 7548	. . . . . . 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 7469	. . . . . . 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 7376	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
11	9, 10	genpelvu 7514	. . . . . 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 411	. . . . 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 296	. . . 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 4009	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
15	14	rexbidv 2478	. . . . . . . . . . . . . . 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 5520	. . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . 18 ⊢ Q ∈ V
18	17	rabex 4149	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
19	17	rabex 4149	. . . . . . . . . . . . . . . . 17 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
20	18, 19	op2nd 6150	. . . . . . . . . . . . . . . 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 2198	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
22	15, 21	elrab2 2898	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
23	22	biimpi 120	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐿) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
24	23	adantr 276	. . . . . . . . . . . 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 277	. . . . . . . . . . 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 276	. . . . . . . . . 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 112	. . . . . . . . 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 2742	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
29		breq2 4009	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑡 → (𝑆 <Q 𝑢 ↔ 𝑆 <Q 𝑡))
30		ltnqex 7550	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 7551	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V
32	30, 31	op2nd 6150	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = {𝑢 ∣ 𝑆 <Q 𝑢}
33	28, 29, 32	elab2 2887	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34		ltrelnq 7366	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
35	34	brel 4680	. . . . . . . . . . . . 13 ⊢ (𝑆 <Q 𝑡 → (𝑆 ∈ Q ∧ 𝑡 ∈ Q))
36	33, 35	sylbi 121	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → (𝑆 ∈ Q ∧ 𝑡 ∈ Q))
37	36	simprd 114	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → 𝑡 ∈ Q)
38	37	ad2antll 491	. . . . . . . . . 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 276	. . . . . . . . 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 7376	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q) → (𝑠 +Q 𝑡) ∈ Q)
41	27, 39, 40	syl2anc 411	. . . . . . . 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 2240	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
43	42	adantl 277	. . . . . . . 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 167	. . . . . . 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 114	. . . . . . . . . 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 5517	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
47		opeq1 3780	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → ⟨𝑗, 1o⟩ = ⟨𝑚, 1o⟩)
48	47	eceq1d 6573	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑚, 1o⟩] ~Q )
49	48	fveq2d 5521	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑚, 1o⟩] ~Q ))
50	46, 49	oveq12d 5895	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
51	50	breq1d 4015	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ↔ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠))
52	51	cbvrexv 2706	. . . . . . . . . 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 122	. . . . . . . . 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 120	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → 𝑆 <Q 𝑡)
55	54	ad2antll 491	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . . 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	ffvelcdmd 5654	. . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ N → [⟨𝑚, 1o⟩] ~Q ∈ Q)
64		recclnq 7393	. . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
67	62, 65, 66	syl2anc 411	. . . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . 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 147	. . . . . . . . . . . . 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 110	. . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑧 <Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q (𝑣 +Q 𝑤)))
73	72	adantl 277	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 +Q 𝑤) = (𝑤 +Q 𝑧))
76	75	adantl 277	. . . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . 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 147	. . . . . . . . . . . . 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 7399	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
80	79, 34	sotri 5026	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 534	. . . . . . . . . . . 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 4033	. . . . . . . . . . 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 115	. . . . . . . . . 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 2579	. . . . . . . . 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 5892	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆))
88	87	breq1d 4015	. . . . . . . . 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 2706	. . . . . . . 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 134	. . . . . . 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 4009	. . . . . . . . 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 2478	. . . . . . . 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 2895	. . . . . . 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 417	. . . . . 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 115	. . . . 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 2602	. . . 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 115	. 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 3163	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 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3131  ⟨cop 3597   class class class wbr 4005  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  2^nd c2nd 6142  1_oc1o 6412  [cec 6535  Ncnpi 7273   <_N clti 7276   ~_Q ceq 7280  Qcnq 7281   +_Q cplq 7283  *_Qcrq 7285   <_Q cltq 7286  Pcnp 7292   +_P cpp 7294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iplp 7469

This theorem is referenced by:  caucvgprlemladdrl  7679