Theorem eroveu 8375

Description: Lemma for erov 8377 and eroprf 8378. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
eropr.1	⊢ 𝐽 = (𝐴 / 𝑅)
eropr.2	⊢ 𝐾 = (𝐵 / 𝑆)
eropr.3	⊢ (𝜑 → 𝑇 ∈ 𝑍)
eropr.4	⊢ (𝜑 → 𝑅 Er 𝑈)
eropr.5	⊢ (𝜑 → 𝑆 Er 𝑉)
eropr.6	⊢ (𝜑 → 𝑇 Er 𝑊)
eropr.7	⊢ (𝜑 → 𝐴 ⊆ 𝑈)
eropr.8	⊢ (𝜑 → 𝐵 ⊆ 𝑉)
eropr.9	⊢ (𝜑 → 𝐶 ⊆ 𝑊)
eropr.10	⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
eropr.11	⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))

Assertion

Ref	Expression
eroveu	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝐽,𝑝,𝑞,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝐾,𝑝,𝑞,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧

Allowed substitution hints:   𝐶(𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝐾(𝑢,𝑡,𝑠,𝑟)   𝑉(𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑊(𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑍(𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem eroveu

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 8333	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐴 / 𝑅) → ∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅)
2		eropr.1	. . . . . . . 8 ⊢ 𝐽 = (𝐴 / 𝑅)
3	1, 2	eleq2s 2908	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → ∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅)
4		elqsi 8333	. . . . . . . 8 ⊢ (𝑌 ∈ (𝐵 / 𝑆) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆)
5		eropr.2	. . . . . . . 8 ⊢ 𝐾 = (𝐵 / 𝑆)
6	4, 5	eleq2s 2908	. . . . . . 7 ⊢ (𝑌 ∈ 𝐾 → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆)
7	3, 6	anim12i 615	. . . . . 6 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾) → (∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆))
8	7	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆))
9		reeanv 3320	. . . . 5 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆))
10	8, 9	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆))
11		eropr.3	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑍)
12	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → 𝑇 ∈ 𝑍)
13		ecexg 8276	. . . . . . 7 ⊢ (𝑇 ∈ 𝑍 → [(𝑝 + 𝑞)]𝑇 ∈ V)
14		elisset 3452	. . . . . . 7 ⊢ ([(𝑝 + 𝑞)]𝑇 ∈ V → ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)
15	12, 13, 14	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)
16	15	biantrud 535	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)))
17	16	2rexbidv 3259	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)))
18	10, 17	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇))
19		19.42v 1954	. . . . . . . 8 ⊢ (∃𝑧((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇))
20	19	bicomi 227	. . . . . . 7 ⊢ (((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑧((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
21	20	rexbii 3210	. . . . . 6 ⊢ (∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑞 ∈ 𝐵 ∃𝑧((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
22		rexcom4 3212	. . . . . 6 ⊢ (∃𝑞 ∈ 𝐵 ∃𝑧((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑧∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
23	21, 22	bitri 278	. . . . 5 ⊢ (∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑧∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
24	23	rexbii 3210	. . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑧∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
25		rexcom4 3212	. . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑧∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
26	24, 25	bitri 278	. . 3 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
27	18, 26	sylib 221	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
28		reeanv 3320	. . . . . 6 ⊢ (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) ↔ (∃𝑟 ∈ 𝐴 ∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑠 ∈ 𝐴 ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
29		eceq1 8310	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑟 → [𝑝]𝑅 = [𝑟]𝑅)
30	29	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑝 = 𝑟 → (𝑋 = [𝑝]𝑅 ↔ 𝑋 = [𝑟]𝑅))
31	30	anbi1d 632	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆)))
32		oveq1 7142	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑟 → (𝑝 + 𝑞) = (𝑟 + 𝑞))
33	32	eceq1d 8311	. . . . . . . . . 10 ⊢ (𝑝 = 𝑟 → [(𝑝 + 𝑞)]𝑇 = [(𝑟 + 𝑞)]𝑇)
34	33	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑧 = [(𝑟 + 𝑞)]𝑇))
35	31, 34	anbi12d 633	. . . . . . . 8 ⊢ (𝑝 = 𝑟 → (((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑟 + 𝑞)]𝑇)))
36		eceq1 8310	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → [𝑞]𝑆 = [𝑡]𝑆)
37	36	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (𝑌 = [𝑞]𝑆 ↔ 𝑌 = [𝑡]𝑆))
38	37	anbi2d 631	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆)))
39		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → (𝑟 + 𝑞) = (𝑟 + 𝑡))
40	39	eceq1d 8311	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → [(𝑟 + 𝑞)]𝑇 = [(𝑟 + 𝑡)]𝑇)
41	40	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → (𝑧 = [(𝑟 + 𝑞)]𝑇 ↔ 𝑧 = [(𝑟 + 𝑡)]𝑇))
42	38, 41	anbi12d 633	. . . . . . . 8 ⊢ (𝑞 = 𝑡 → (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑟 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇)))
43	35, 42	cbvrex2vw 3409	. . . . . . 7 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑟 ∈ 𝐴 ∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇))
44		eceq1 8310	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → [𝑝]𝑅 = [𝑠]𝑅)
45	44	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑝 = 𝑠 → (𝑋 = [𝑝]𝑅 ↔ 𝑋 = [𝑠]𝑅))
46	45	anbi1d 632	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆)))
47		oveq1 7142	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝑝 + 𝑞) = (𝑠 + 𝑞))
48	47	eceq1d 8311	. . . . . . . . . 10 ⊢ (𝑝 = 𝑠 → [(𝑝 + 𝑞)]𝑇 = [(𝑠 + 𝑞)]𝑇)
49	48	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → (𝑤 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑠 + 𝑞)]𝑇))
50	46, 49	anbi12d 633	. . . . . . . 8 ⊢ (𝑝 = 𝑠 → (((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑠 + 𝑞)]𝑇)))
51		eceq1 8310	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑢 → [𝑞]𝑆 = [𝑢]𝑆)
52	51	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑞 = 𝑢 → (𝑌 = [𝑞]𝑆 ↔ 𝑌 = [𝑢]𝑆))
53	52	anbi2d 631	. . . . . . . . 9 ⊢ (𝑞 = 𝑢 → ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆)))
54		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑢 → (𝑠 + 𝑞) = (𝑠 + 𝑢))
55	54	eceq1d 8311	. . . . . . . . . 10 ⊢ (𝑞 = 𝑢 → [(𝑠 + 𝑞)]𝑇 = [(𝑠 + 𝑢)]𝑇)
56	55	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑞 = 𝑢 → (𝑤 = [(𝑠 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑠 + 𝑢)]𝑇))
57	53, 56	anbi12d 633	. . . . . . . 8 ⊢ (𝑞 = 𝑢 → (((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑠 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
58	50, 57	cbvrex2vw 3409	. . . . . . 7 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑠 ∈ 𝐴 ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇))
59	43, 58	anbi12i 629	. . . . . 6 ⊢ ((∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)) ↔ (∃𝑟 ∈ 𝐴 ∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑠 ∈ 𝐴 ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
60	28, 59	bitr4i 281	. . . . 5 ⊢ (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) ↔ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)))
61		reeanv 3320	. . . . . . 7 ⊢ (∃𝑡 ∈ 𝐵 ∃𝑢 ∈ 𝐵 (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) ↔ (∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
62		eropr.11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
63		eropr.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 Er 𝑈)
64	63	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑅 Er 𝑈)
65		eropr.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
66	65	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐴 ⊆ 𝑈)
67		simprll 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑟 ∈ 𝐴)
68	66, 67	sseldd 3916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑟 ∈ 𝑈)
69	64, 68	erth 8321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑟𝑅𝑠 ↔ [𝑟]𝑅 = [𝑠]𝑅))
70		eropr.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 Er 𝑉)
71	70	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑆 Er 𝑉)
72		eropr.8	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
73	72	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐵 ⊆ 𝑉)
74		simprrl 780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑡 ∈ 𝐵)
75	73, 74	sseldd 3916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑡 ∈ 𝑉)
76	71, 75	erth 8321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑡𝑆𝑢 ↔ [𝑡]𝑆 = [𝑢]𝑆))
77	69, 76	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) ↔ ([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆)))
78		eropr.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 Er 𝑊)
79	78	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑇 Er 𝑊)
80		eropr.9	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ⊆ 𝑊)
81	80	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐶 ⊆ 𝑊)
82		eropr.10	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
83	82	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → + :(𝐴 × 𝐵)⟶𝐶)
84	83, 67, 74	fovrnd 7300	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑟 + 𝑡) ∈ 𝐶)
85	81, 84	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑟 + 𝑡) ∈ 𝑊)
86	79, 85	erth 8321	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟 + 𝑡)𝑇(𝑠 + 𝑢) ↔ [(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇))
87	62, 77, 86	3imtr3d 296	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) → [(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇))
88		eqeq2 2810	. . . . . . . . . . . . . 14 ⊢ (𝑤 = [(𝑠 + 𝑢)]𝑇 → ([(𝑟 + 𝑡)]𝑇 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇))
89	88	biimprcd 253	. . . . . . . . . . . . 13 ⊢ ([(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇 → (𝑤 = [(𝑠 + 𝑢)]𝑇 → [(𝑟 + 𝑡)]𝑇 = 𝑤))
90	87, 89	syl6 35	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) → (𝑤 = [(𝑠 + 𝑢)]𝑇 → [(𝑟 + 𝑡)]𝑇 = 𝑤)))
91	90	impd 414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) → [(𝑟 + 𝑡)]𝑇 = 𝑤))
92		eqeq1 2802	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = [𝑟]𝑅 → (𝑋 = [𝑠]𝑅 ↔ [𝑟]𝑅 = [𝑠]𝑅))
93		eqeq1 2802	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = [𝑡]𝑆 → (𝑌 = [𝑢]𝑆 ↔ [𝑡]𝑆 = [𝑢]𝑆))
94	92, 93	bi2anan9 638	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) → ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ↔ ([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆)))
95	94	anbi1d 632	. . . . . . . . . . . . 13 ⊢ ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) → (((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) ↔ (([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
96	95	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) → (((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) ↔ (([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)))
97		eqeq1 2802	. . . . . . . . . . . . 13 ⊢ (𝑧 = [(𝑟 + 𝑡)]𝑇 → (𝑧 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = 𝑤))
98	97	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) → (𝑧 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = 𝑤))
99	96, 98	imbi12d 348	. . . . . . . . . . 11 ⊢ (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) → ((((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) → 𝑧 = 𝑤) ↔ ((([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) → [(𝑟 + 𝑡)]𝑇 = 𝑤)))
100	91, 99	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) → (((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇) → 𝑧 = 𝑤)))
101	100	impd 414	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) → 𝑧 = 𝑤))
102	101	anassrs 471	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → ((((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) → 𝑧 = 𝑤))
103	102	rexlimdvva 3253	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) → (∃𝑡 ∈ 𝐵 ∃𝑢 ∈ 𝐵 (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) → 𝑧 = 𝑤))
104	61, 103	syl5bir 246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) → ((∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) → 𝑧 = 𝑤))
105	104	rexlimdvva 3253	. . . . 5 ⊢ (𝜑 → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (∃𝑡 ∈ 𝐵 ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) ∧ ∃𝑢 ∈ 𝐵 ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ∧ 𝑤 = [(𝑠 + 𝑢)]𝑇)) → 𝑧 = 𝑤))
106	60, 105	syl5bir 246	. . . 4 ⊢ (𝜑 → ((∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)) → 𝑧 = 𝑤))
107	106	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ((∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)) → 𝑧 = 𝑤))
108	107	alrimivv 1929	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∀𝑧∀𝑤((∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)) → 𝑧 = 𝑤))
109		eqeq1 2802	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑝 + 𝑞)]𝑇))
110	109	anbi2d 631	. . . 4 ⊢ (𝑧 = 𝑤 → (((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)))
111	110	2rexbidv 3259	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)))
112	111	eu4 2676	. 2 ⊢ (∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (∃𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∀𝑧∀𝑤((∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑤 = [(𝑝 + 𝑞)]𝑇)) → 𝑧 = 𝑤)))
113	27, 108, 112	sylanbrc 586	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   × cxp 5517  ⟶wf 6320  (class class class)co 7135   Er wer 8269  [cec 8270   / cqs 8271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-er 8272  df-ec 8274  df-qs 8278

This theorem is referenced by:  erovlem  8376  erov  8377  eroprf  8378