Theorem cauappcvgprlemladdfu 7974

Description: Lemma for cauappcvgprlemladd 7978. 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 7973	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ P)
6		cauappcvgprlemladd.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Q)
7		nqprlu 7867	. . . . . . 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 7788	. . . . . . 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 7695	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
11	9, 10	genpelvu 7833	. . . . . 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 4115	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
15	14	rexbidv 2545	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑠 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠))
16	4	fveq2i 5675	. . . . . . . . . . . . . . . 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 7683	. . . . . . . . . . . . . . . . . 18 ⊢ Q ∈ V
18	17	rabex 4258	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
19	17	rabex 4258	. . . . . . . . . . . . . . . . 17 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
20	18, 19	op2nd 6343	. . . . . . . . . . . . . . . 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 2255	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
22	15, 21	elrab2 2978	. . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
29		breq2 4115	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑡 → (𝑆 <Q 𝑢 ↔ 𝑆 <Q 𝑡))
30		ltnqex 7869	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V
31		gtnqex 7870	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V
32	30, 31	op2nd 6343	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = {𝑢 ∣ 𝑆 <Q 𝑢}
33	28, 29, 32	elab2 2967	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34		ltrelnq 7685	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
35	34	brel 4804	. . . . . . . . . . . . 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 7695	. . . . . . . . 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 2297	. . . . . . . . 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 491	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 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 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑡 ∈ Q)
52	1	ad5antr 496	. . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . 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 5815	. . . . . . . . . . . . . . 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 7695	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑞) ∈ Q ∧ 𝑞 ∈ Q) → ((𝐹‘𝑞) +Q 𝑞) ∈ Q)
56	54, 53, 55	syl2anc 411	. . . . . . . . . . . . . 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 7720	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 7720	. . . . . . . . . . . . . . 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 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)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑠 ∈ Q)
64		addcomnqg 7701	. . . . . . . . . . . . . . 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 6226	. . . . . . . . . . . . 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 7718	. . . . . . . . . . . . 13 ⊢ <Q Or Q
69	68, 34	sotri 5160	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 536	. . . . . . . . . . 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 4139	. . . . . . . . . 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 2646	. . . . . . . 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 4115	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → ((((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
77	76	rexbidv 2545	. . . . . . . 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 4258	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)} ∈ V
79	17	rabex 4258	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
80	78, 79	op2nd 6343	. . . . . . . 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 2978	. . . . . . 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 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 ∈ (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 2670	. . . 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 3246	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}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  {crab 2526   ⊆ wss 3213  ⟨cop 3694   class class class wbr 4111  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052  2^nd c2nd 6335  Qcnq 7600   +_Q cplq 7602   <_Q cltq 7605  Pcnp 7611   +_P cpp 7613

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673  df-inp 7786  df-iplp 7788

This theorem is referenced by:  cauappcvgprlemladdrl  7977  cauappcvgprlemladd  7978