Theorem cauappcvgprlemladdfu 8022

Description: Lemma for cauappcvgprlemladd 8026. The forward subset relationship for the upper 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
cauappcvgprlemladdfu	⊢ (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{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 cauappcvgprlemladdfu

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 8021	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		cauappcvgprlemladd.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‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
15	14	rexbidv 2551	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑠 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
16	4	fveq2i 5698	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
17		nqex 7731	. . . . . . . . . . . . . . . . . 18 ⊢ Q ∈ V
18	17	rabex 4280	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
19	17	rabex 4280	. . . . . . . . . . . . . . . . 17 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
20	18, 19	op2nd 6381	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
21	16, 20	eqtri 2259	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
22	15, 21	elrab2 2985	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
23	22	biimpi 120	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐿) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
24	23	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) → (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 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 ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 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 ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 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	. . . . . . . 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)) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)
46	33	biimpi 120	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) → 𝑆 <Q 𝑡)
47	46	ad2antll 495	. . . . . . . . . . . . . . 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}⟩))) → 𝑆 <Q 𝑡)
48	47	adantr 276	. . . . . . . . . . . . . 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)) → 𝑆 <Q 𝑡)
49	48	ad2antrr 492	. . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 <Q 𝑡)
50	6	ad5antr 500	. . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 ∈ Q)
51	39	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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑡 ∈ Q)
52	1	ad5antr 500	. . . . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝐹:Q⟶Q)
53		simplr 533	. . . . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑞 ∈ Q)
54	52, 53	ffvelcdmd 5844	. . . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (𝐹‘𝑞) ∈ Q)
55		addclnq 7743	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑞) ∈ Q ∧ 𝑞 ∈ Q) → ((𝐹‘𝑞) +Q 𝑞) ∈ Q)
56	54, 53, 55	syl2anc 415	. . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹‘𝑞) +Q 𝑞) ∈ Q)
57		ltanqg 7768	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Q ∧ 𝑡 ∈ Q ∧ ((𝐹‘𝑞) +Q 𝑞) ∈ Q) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡)))
58	50, 51, 56, 57	syl3anc 1278	. . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡)))
59	49, 58	mpbid 147	. . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡))
60		simpr 110	. . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)
61		ltanqg 7768	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
62	61	adantl 277	. . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
63	27	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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑠 ∈ Q)
64		addcomnqg 7749	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
65	64	adantl 277	. . . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
66	62, 56, 63, 51, 65	caovord2d 6259	. . . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)))
67	60, 66	mpbid 147	. . . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡))
68		ltsonq 7766	. . . . . . . . . . . . 13 ⊢ <Q Or Q
69	68, 34	sotri 5183	. . . . . . . . . . . 12 ⊢ (((((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) ∧ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (𝑠 +Q 𝑡))
70	59, 67, 69	syl2anc 415	. . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q (𝑠 +Q 𝑡))
71		simpllr 540	. . . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
72	70, 71	breqtrrd 4158	. . . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
73	72	ex 115	. . . . . . . . 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)) ∧ 𝑞 ∈ Q) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
74	73	reximdva 2652	. . . . . . . 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)) → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
75	45, 74	mpd 13	. . . . . . 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)) → ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
76		breq2 4134	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → ((((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
77	76	rexbidv 2551	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → (∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢 ↔ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
78	17	rabex 4280	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)} ∈ V
79	17	rabex 4280	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
80	78, 79	op2nd 6381	. . . . . . . 8 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}
81	77, 80	elrab2 2985	. . . . . . 7 ⊢ (𝑟 ∈ (2nd ‘⟨{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 (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
82	44, 75, 81	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 ∈ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
83	82	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 ∈ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
84	83	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 ∈ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
85	13, 84	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
86	85	ex 115	. 2 ⊢ (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
87	86	ssrdv 3254	1 ⊢ (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 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  Qcnq 7648   +_Q cplq 7650   <_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:  cauappcvgprlemladdrl  8025  cauappcvgprlemladd  8026