Theorem eroveu 6616

Description: Lemma for eroprf 6618. (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 6577	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐴 / 𝑅) → ∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅)
2		eropr.1	. . . . . . . 8 ⊢ 𝐽 = (𝐴 / 𝑅)
3	1, 2	eleq2s 2270	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → ∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅)
4		elqsi 6577	. . . . . . . 8 ⊢ (𝑌 ∈ (𝐵 / 𝑆) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆)
5		eropr.2	. . . . . . . 8 ⊢ 𝐾 = (𝐵 / 𝑆)
6	4, 5	eleq2s 2270	. . . . . . 7 ⊢ (𝑌 ∈ 𝐾 → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆)
7	3, 6	anim12i 338	. . . . . 6 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾) → (∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆))
8	7	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (∃𝑝 ∈ 𝐴 𝑋 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑌 = [𝑞]𝑆))
9		reeanv 2644	. . . . 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 134	. . . 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 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → 𝑇 ∈ 𝑍)
13		ecexg 6529	. . . . . . 7 ⊢ (𝑇 ∈ 𝑍 → [(𝑝 + 𝑞)]𝑇 ∈ V)
14		elisset 2749	. . . . . . 7 ⊢ ([(𝑝 + 𝑞)]𝑇 ∈ V → ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)
15	12, 13, 14	3syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇)
16	15	biantrud 304	. . . . 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 2500	. . . 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 147	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ ∃𝑧 𝑧 = [(𝑝 + 𝑞)]𝑇))
19		19.42v 1904	. . . . . . . 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 132	. . . . . . 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 2482	. . . . . 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 2758	. . . . . 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 184	. . . . 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 2482	. . . 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 2758	. . . 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 184	. . 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 122	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
28		reeanv 2644	. . . . . 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 6560	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑟 → [𝑝]𝑅 = [𝑟]𝑅)
30	29	eqeq2d 2187	. . . . . . . . . 10 ⊢ (𝑝 = 𝑟 → (𝑋 = [𝑝]𝑅 ↔ 𝑋 = [𝑟]𝑅))
31	30	anbi1d 465	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆)))
32		oveq1 5872	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑟 → (𝑝 + 𝑞) = (𝑟 + 𝑞))
33	32	eceq1d 6561	. . . . . . . . . 10 ⊢ (𝑝 = 𝑟 → [(𝑝 + 𝑞)]𝑇 = [(𝑟 + 𝑞)]𝑇)
34	33	eqeq2d 2187	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑧 = [(𝑟 + 𝑞)]𝑇))
35	31, 34	anbi12d 473	. . . . . . . 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 6560	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → [𝑞]𝑆 = [𝑡]𝑆)
37	36	eqeq2d 2187	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (𝑌 = [𝑞]𝑆 ↔ 𝑌 = [𝑡]𝑆))
38	37	anbi2d 464	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆)))
39		oveq2 5873	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → (𝑟 + 𝑞) = (𝑟 + 𝑡))
40	39	eceq1d 6561	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → [(𝑟 + 𝑞)]𝑇 = [(𝑟 + 𝑡)]𝑇)
41	40	eqeq2d 2187	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → (𝑧 = [(𝑟 + 𝑞)]𝑇 ↔ 𝑧 = [(𝑟 + 𝑡)]𝑇))
42	38, 41	anbi12d 473	. . . . . . . 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	cbvrex2v 2715	. . . . . . 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 6560	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → [𝑝]𝑅 = [𝑠]𝑅)
45	44	eqeq2d 2187	. . . . . . . . . 10 ⊢ (𝑝 = 𝑠 → (𝑋 = [𝑝]𝑅 ↔ 𝑋 = [𝑠]𝑅))
46	45	anbi1d 465	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆)))
47		oveq1 5872	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝑝 + 𝑞) = (𝑠 + 𝑞))
48	47	eceq1d 6561	. . . . . . . . . 10 ⊢ (𝑝 = 𝑠 → [(𝑝 + 𝑞)]𝑇 = [(𝑠 + 𝑞)]𝑇)
49	48	eqeq2d 2187	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → (𝑤 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑠 + 𝑞)]𝑇))
50	46, 49	anbi12d 473	. . . . . . . 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 6560	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑢 → [𝑞]𝑆 = [𝑢]𝑆)
52	51	eqeq2d 2187	. . . . . . . . . 10 ⊢ (𝑞 = 𝑢 → (𝑌 = [𝑞]𝑆 ↔ 𝑌 = [𝑢]𝑆))
53	52	anbi2d 464	. . . . . . . . 9 ⊢ (𝑞 = 𝑢 → ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑞]𝑆) ↔ (𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆)))
54		oveq2 5873	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑢 → (𝑠 + 𝑞) = (𝑠 + 𝑢))
55	54	eceq1d 6561	. . . . . . . . . 10 ⊢ (𝑞 = 𝑢 → [(𝑠 + 𝑞)]𝑇 = [(𝑠 + 𝑢)]𝑇)
56	55	eqeq2d 2187	. . . . . . . . 9 ⊢ (𝑞 = 𝑢 → (𝑤 = [(𝑠 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑠 + 𝑢)]𝑇))
57	53, 56	anbi12d 473	. . . . . . . 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	cbvrex2v 2715	. . . . . . 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 460	. . . . . 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 187	. . . . 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 2644	. . . . . . 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 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑅 Er 𝑈)
65		eropr.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
66	65	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐴 ⊆ 𝑈)
67		simprll 537	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑟 ∈ 𝐴)
68	66, 67	sseldd 3154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑟 ∈ 𝑈)
69	64, 68	erth 6569	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑆 Er 𝑉)
72		eropr.8	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
73	72	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐵 ⊆ 𝑉)
74		simprrl 539	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑡 ∈ 𝐵)
75	73, 74	sseldd 3154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑡 ∈ 𝑉)
76	71, 75	erth 6569	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑡𝑆𝑢 ↔ [𝑡]𝑆 = [𝑢]𝑆))
77	69, 76	anbi12d 473	. . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝑇 Er 𝑊)
80		eropr.9	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ⊆ 𝑊)
81	80	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → 𝐶 ⊆ 𝑊)
82		eropr.10	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
83	82	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → + :(𝐴 × 𝐵)⟶𝐶)
84	83, 67, 74	fovcdmd 6009	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑟 + 𝑡) ∈ 𝐶)
85	81, 84	sseldd 3154	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → (𝑟 + 𝑡) ∈ 𝑊)
86	79, 85	erth 6569	. . . . . . . . . . . . . 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 202	. . . . . . . . . . . . 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 2185	. . . . . . . . . . . . . 14 ⊢ (𝑤 = [(𝑠 + 𝑢)]𝑇 → ([(𝑟 + 𝑡)]𝑇 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇))
89	88	biimprcd 160	. . . . . . . . . . . . 13 ⊢ ([(𝑟 + 𝑡)]𝑇 = [(𝑠 + 𝑢)]𝑇 → (𝑤 = [(𝑠 + 𝑢)]𝑇 → [(𝑟 + 𝑡)]𝑇 = 𝑤))
90	87, 89	syl6 33	. . . . . . . . . . . 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 254	. . . . . . . . . . 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 2182	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = [𝑟]𝑅 → (𝑋 = [𝑠]𝑅 ↔ [𝑟]𝑅 = [𝑠]𝑅))
93		eqeq1 2182	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = [𝑡]𝑆 → (𝑌 = [𝑢]𝑆 ↔ [𝑡]𝑆 = [𝑢]𝑆))
94	92, 93	bi2anan9 606	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) → ((𝑋 = [𝑠]𝑅 ∧ 𝑌 = [𝑢]𝑆) ↔ ([𝑟]𝑅 = [𝑠]𝑅 ∧ [𝑡]𝑆 = [𝑢]𝑆)))
95	94	anbi1d 465	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 2182	. . . . . . . . . . . . 13 ⊢ (𝑧 = [(𝑟 + 𝑡)]𝑇 → (𝑧 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = 𝑤))
98	97	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝑋 = [𝑟]𝑅 ∧ 𝑌 = [𝑡]𝑆) ∧ 𝑧 = [(𝑟 + 𝑡)]𝑇) → (𝑧 = 𝑤 ↔ [(𝑟 + 𝑡)]𝑇 = 𝑤))
99	96, 98	imbi12d 234	. . . . . . . . . . 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 157	. . . . . . . . . 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 254	. . . . . . . . 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 400	. . . . . . . 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 2600	. . . . . . 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 153	. . . . . 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 2600	. . . . 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 153	. . . 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 276	. . 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 1873	. 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 2182	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ 𝑤 = [(𝑝 + 𝑞)]𝑇))
110	109	anbi2d 464	. . . 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 2500	. . 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 2086	. 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 417	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   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1490  ∃!weu 2024   ∈ wcel 2146  ∃wrex 2454  Vcvv 2735   ⊆ wss 3127   class class class wbr 3998   × cxp 4618  ⟶wf 5204  (class class class)co 5865   Er wer 6522  [cec 6523   / cqs 6524

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-er 6525  df-ec 6527  df-qs 6531

This theorem is referenced by:  erovlem  6617  eroprf  6618