Theorem cauappcvgprlemladdfl 7604

Description: Lemma for cauappcvgprlemladd 7607. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
cauappcvgprlemladd.s	⊢ (𝜑 → 𝑆 ∈ Q)

Assertion

Ref	Expression
cauappcvgprlemladdfl	⊢ (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑙,𝑢,𝑝,𝑞   𝑆,𝑙,𝑞,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝑆(𝑝)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemladdfl

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

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:Q⟶Q)
2		cauappcvgpr.app	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
3		cauappcvgpr.bnd	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		cauappcvgpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
5	1, 2, 3, 4	cauappcvgprlemcl 7602	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		cauappcvgprlemladd.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Q)
7		nqprlu 7496	. . . . . . 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 7417	. . . . . . 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 7324	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
11	9, 10	genpelvl 7461	. . . . . 6 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
12	5, 8, 11	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
13	12	biimpa 294	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → ∃𝑠 ∈ (1st ‘𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
14		oveq1 5857	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
15	14	breq1d 3997	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
16	15	rexbidv 2471	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
17	4	fveq2i 5497	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
18		nqex 7312	. . . . . . . . . . . . . . . . 17 ⊢ Q ∈ V
19	18	rabex 4131	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
20	18	rabex 4131	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
21	19, 20	op1st 6122	. . . . . . . . . . . . . . 15 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
22	17, 21	eqtri 2191	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
23	16, 22	elrab2 2889	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
24	23	biimpi 119	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (1st ‘𝐿) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
25	24	ad2antrl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
26	25	adantr 274	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
27	26	simpld 111	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 ∈ Q)
28		vex 2733	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
29		breq1 3990	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑡 → (𝑙 <Q 𝑆 ↔ 𝑡 <Q 𝑆))
30		ltnqex 7498	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 7499	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V
32	30, 31	op1st 6122	. . . . . . . . . . . . . . 15 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝑆}
33	28, 29, 32	elab2 2878	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝑆)
34	33	biimpi 119	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → 𝑡 <Q 𝑆)
35	34	ad2antll 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → 𝑡 <Q 𝑆)
36	35	adantr 274	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 <Q 𝑆)
37		ltrelnq 7314	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
38	37	brel 4661	. . . . . . . . . . 11 ⊢ (𝑡 <Q 𝑆 → (𝑡 ∈ Q ∧ 𝑆 ∈ Q))
39	36, 38	syl 14	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑡 ∈ Q ∧ 𝑆 ∈ Q))
40	39	simpld 111	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 ∈ Q)
41		addclnq 7324	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q) → (𝑠 +Q 𝑡) ∈ Q)
42	27, 40, 41	syl2anc 409	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈ Q)
43		eleq1 2233	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
44	43	adantl 275	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
45	42, 44	mpbird 166	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ Q)
46	26	simprd 113	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
47	27	ad2antrr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑠 ∈ Q)
48		simplr 525	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑞 ∈ Q)
49	40	ad2antrr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑡 ∈ Q)
50		addcomnqg 7330	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
51	50	adantl 275	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
52		addassnqg 7331	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
53	52	adantl 275	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
54	47, 48, 49, 51, 53	caov32d 6030	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) = ((𝑠 +Q 𝑡) +Q 𝑞))
55		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
56	35	ad2antrr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑡 <Q 𝑆)
57	37	brel 4661	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) → ((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹‘𝑞) ∈ Q))
58		lt2addnq 7353	. . . . . . . . . . . . . . 15 ⊢ ((((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹‘𝑞) ∈ Q) ∧ (𝑡 ∈ Q ∧ 𝑆 ∈ Q)) → (((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ∧ 𝑡 <Q 𝑆) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹‘𝑞) +Q 𝑆)))
59	57, 39, 58	syl2anr 288	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → (((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ∧ 𝑡 <Q 𝑆) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹‘𝑞) +Q 𝑆)))
60	55, 56, 59	mp2and 431	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹‘𝑞) +Q 𝑆))
61	60	adantlr 474	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹‘𝑞) +Q 𝑆))
62	54, 61	eqbrtrrd 4011	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆))
63		oveq1 5857	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
64	63	breq1d 3997	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠 +Q 𝑡) → ((𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
65	64	ad3antlr 490	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → ((𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
66	62, 65	mpbird 166	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆))
67	66	ex 114	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) → ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) → (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
68	67	reximdva 2572	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
69	46, 68	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆))
70		oveq1 5857	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
71	70	breq1d 3997	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
72	71	rexbidv 2471	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
73	18	rabex 4131	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)} ∈ V
74	18	rabex 4131	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
75	73, 74	op1st 6122	. . . . . . . 8 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}
76	72, 75	elrab2 2889	. . . . . . 7 ⊢ (𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
77	45, 69, 76	sylanbrc 415	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
78	77	ex 114	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
79	78	rexlimdvva 2595	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → (∃𝑠 ∈ (1st ‘𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
80	13, 79	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → 𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
81	80	ex 114	. 2 ⊢ (𝜑 → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
82	81	ssrdv 3153	1 ⊢ (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  {crab 2452   ⊆ wss 3121  ⟨cop 3584   class class class wbr 3987  ⟶wf 5192  ‘cfv 5196  (class class class)co 5850  1^st c1st 6114  Qcnq 7229   +_Q cplq 7231   <_Q cltq 7234  Pcnp 7240   +_P cpp 7242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-inp 7415  df-iplp 7417

This theorem is referenced by:  cauappcvgprlemladdru  7605  cauappcvgprlemladd  7607