Theorem cauappcvgprlemladdfl 7656

Description: Lemma for cauappcvgprlemladd 7659. 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 7654	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		cauappcvgprlemladd.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	genpelvl 7513	. . . . . 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 411	. . . . 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 296	. . . 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 5884	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
15	14	breq1d 4015	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
16	15	rexbidv 2478	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
17	4	fveq2i 5520	. . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . 17 ⊢ Q ∈ V
19	18	rabex 4149	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
20	18	rabex 4149	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
21	19, 20	op1st 6149	. . . . . . . . . . . . . . 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 2198	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
23	16, 22	elrab2 2898	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
24	23	biimpi 120	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (1st ‘𝐿) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
25	24	ad2antrl 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}⟩))) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
26	25	adantr 276	. . . . . . . . . 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 112	. . . . . . . . 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 2742	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
29		breq1 4008	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑡 → (𝑙 <Q 𝑆 ↔ 𝑡 <Q 𝑆))
30		ltnqex 7550	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 7551	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V
32	30, 31	op1st 6149	. . . . . . . . . . . . . . 15 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝑆}
33	28, 29, 32	elab2 2887	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝑆)
34	33	biimpi 120	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → 𝑡 <Q 𝑆)
35	34	ad2antll 491	. . . . . . . . . . . 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 276	. . . . . . . . . . 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 7366	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
38	37	brel 4680	. . . . . . . . . . 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 112	. . . . . . . . 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 7376	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q) → (𝑠 +Q 𝑡) ∈ Q)
42	27, 40, 41	syl2anc 411	. . . . . . . 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 2240	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈ Q ↔ (𝑠 +Q 𝑡) ∈ Q))
44	43	adantl 277	. . . . . . . 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 167	. . . . . . 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 114	. . . . . . . 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 488	. . . . . . . . . . . . 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 528	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
51	50	adantl 277	. . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
53	52	adantl 277	. . . . . . . . . . . . 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 6057	. . . . . . . . . . . 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 110	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 4680	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) → ((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹‘𝑞) ∈ Q))
58		lt2addnq 7405	. . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . 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 433	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 4029	. . . . . . . . . . 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 5884	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
64	63	breq1d 4015	. . . . . . . . . . . 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 493	. . . . . . . . . . 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 167	. . . . . . . . . 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 115	. . . . . . . . 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 2579	. . . . . . . 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 5884	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
71	70	breq1d 4015	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)))
72	71	rexbidv 2478	. . . . . . . 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 4149	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)} ∈ V
74	18	rabex 4149	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
75	73, 74	op1st 6149	. . . . . . . 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 2898	. . . . . . 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 417	. . . . . 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 115	. . . . 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 2602	. . . 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 115	. 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 3163	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 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  1^st c1st 6141  Qcnq 7281   +_Q cplq 7283   <_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:  cauappcvgprlemladdru  7657  cauappcvgprlemladd  7659