Theorem caucvgprlemladdfu 8045

Description: Lemma for caucvgpr 8050. 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 8044	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		caucvgprlemladd.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Q)
7		nqprlu 7915	. . . . . . 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 7836	. . . . . . 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 7743	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
11	9, 10	genpelvu 7881	. . . . . 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 415	. . . . 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 4134	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
15	14	rexbidv 2551	. . . . . . . . . . . . . . 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 5698	. . . . . . . . . . . . . . . 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 7731	. . . . . . . . . . . . . . . . . 18 ⊢ Q ∈ V
18	17	rabex 4280	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
19	17	rabex 4280	. . . . . . . . . . . . . . . . 17 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
20	18, 19	op2nd 6381	. . . . . . . . . . . . . . . 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 2259	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
22	15, 21	elrab2 2985	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
29		breq2 4134	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑡 → (𝑆 <Q 𝑢 ↔ 𝑆 <Q 𝑡))
30		ltnqex 7917	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 7918	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V
32	30, 31	op2nd 6381	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = {𝑢 ∣ 𝑆 <Q 𝑢}
33	28, 29, 32	elab2 2974	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34		ltrelnq 7733	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
35	34	brel 4827	. . . . . . . . . . . . 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 495	. . . . . . . . . 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 7743	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q) → (𝑠 +Q 𝑡) ∈ Q)
41	27, 39, 40	syl2anc 415	. . . . . . . 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 2301	. . . . . . . . 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 5695	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
47		opeq1 3904	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → ⟨𝑗, 1o⟩ = ⟨𝑚, 1o⟩)
48	47	eceq1d 6843	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑚, 1o⟩] ~Q )
49	48	fveq2d 5699	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑚, 1o⟩] ~Q ))
50	46, 49	oveq12d 6103	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
51	50	breq1d 4140	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ↔ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠))
52	51	cbvrexv 2787	. . . . . . . . . 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 495	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . . . 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 5844	. . . . . . . . . . . . . . . 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 7790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ N → [⟨𝑚, 1o⟩] ~Q ∈ Q)
64		recclnq 7760	. . . . . . . . . . . . . . . . 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 7743	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
67	62, 65, 66	syl2anc 415	. . . . . . . . . . . . . . 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 7768	. . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . 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 7768	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . 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 7749	. . . . . . . . . . . . . . . 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 6259	. . . . . . . . . . . . . 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 7766	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
80	79, 34	sotri 5183	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 4158	. . . . . . . . . . 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 2652	. . . . . . . . 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 6100	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆))
88	87	breq1d 4140	. . . . . . . . 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 2787	. . . . . . . 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 4134	. . . . . . . . 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 2551	. . . . . . . 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 2982	. . . . . . 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 421	. . . . . 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 2676	. . . 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 3254	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})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4130  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  2^nd c2nd 6373  1_oc1o 6680  [cec 6805  Ncnpi 7640   <_N clti 7643   ~_Q ceq 7647  Qcnq 7648   +_Q cplq 7650  *_Qcrq 7652   <_Q cltq 7653  Pcnp 7659   +_P cpp 7661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7672  df-pli 7673  df-mi 7674  df-lti 7675  df-plpq 7712  df-mpq 7713  df-enq 7715  df-nqqs 7716  df-plqqs 7717  df-mqqs 7718  df-1nqqs 7719  df-rq 7720  df-ltnqqs 7721  df-inp 7834  df-iplp 7836

This theorem is used by:  caucvgprlemladdrl  8046