Theorem erov 8753

Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-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)))
eropr.12	⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
eropr.13	⊢ (𝜑 → 𝑅 ∈ 𝑋)
eropr.14	⊢ (𝜑 → 𝑆 ∈ 𝑌)

Assertion

Ref	Expression
erov	⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)

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

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

Proof of Theorem erov

Step	Hyp	Ref	Expression
1		eropr.1	. . . . 5 ⊢ 𝐽 = (𝐴 / 𝑅)
2		eropr.2	. . . . 5 ⊢ 𝐾 = (𝐵 / 𝑆)
3		eropr.3	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑍)
4		eropr.4	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝑈)
5		eropr.5	. . . . 5 ⊢ (𝜑 → 𝑆 Er 𝑉)
6		eropr.6	. . . . 5 ⊢ (𝜑 → 𝑇 Er 𝑊)
7		eropr.7	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
8		eropr.8	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
9		eropr.9	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝑊)
10		eropr.10	. . . . 5 ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
11		eropr.11	. . . . 5 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
12		eropr.12	. . . . 5 ⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	erovlem 8752	. . . 4 ⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
14	13	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
15		simprl 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → 𝑥 = [𝑃]𝑅)
16	15	eqeq1d 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (𝑥 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑝]𝑅))
17		simprr 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → 𝑦 = [𝑄]𝑆)
18	17	eqeq1d 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (𝑦 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑞]𝑆))
19	16, 18	anbi12d 631	. . . . . 6 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
20	19	anbi1d 630	. . . . 5 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
21	20	2rexbidv 3213	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
22	21	iotabidv 6480	. . 3 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
23		eropr.13	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
24		ecelqsg 8711	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ (𝐴 / 𝑅))
25	24, 1	eleqtrrdi 2849	. . . . 5 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ 𝐽)
26	23, 25	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ 𝐽)
27	26	3adant3 1132	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [𝑃]𝑅 ∈ 𝐽)
28		eropr.14	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑌)
29		ecelqsg 8711	. . . . . 6 ⊢ ((𝑆 ∈ 𝑌 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ (𝐵 / 𝑆))
30	29, 2	eleqtrrdi 2849	. . . . 5 ⊢ ((𝑆 ∈ 𝑌 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾)
31	28, 30	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾)
32	31	3adant2 1131	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾)
33		iotaex 6469	. . . 4 ⊢ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V
34	33	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V)
35	14, 22, 27, 32, 34	ovmpod 7507	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
36		eqid 2736	. . . . . . 7 ⊢ [𝑃]𝑅 = [𝑃]𝑅
37		eqid 2736	. . . . . . 7 ⊢ [𝑄]𝑆 = [𝑄]𝑆
38	36, 37	pm3.2i 471	. . . . . 6 ⊢ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)
39		eqid 2736	. . . . . 6 ⊢ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇
40	38, 39	pm3.2i 471	. . . . 5 ⊢ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)
41		eceq1 8686	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → [𝑝]𝑅 = [𝑃]𝑅)
42	41	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ([𝑃]𝑅 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑃]𝑅))
43	42	anbi1d 630	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
44		oveq1 7364	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 + 𝑞) = (𝑃 + 𝑞))
45	44	eceq1d 8687	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → [(𝑝 + 𝑞)]𝑇 = [(𝑃 + 𝑞)]𝑇)
46	45	eqeq2d 2747	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ([(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇))
47	43, 46	anbi12d 631	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇)))
48		eceq1 8686	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → [𝑞]𝑆 = [𝑄]𝑆)
49	48	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ([𝑄]𝑆 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑄]𝑆))
50	49	anbi2d 629	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)))
51		oveq2 7365	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑃 + 𝑞) = (𝑃 + 𝑄))
52	51	eceq1d 8687	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → [(𝑃 + 𝑞)]𝑇 = [(𝑃 + 𝑄)]𝑇)
53	52	eqeq2d 2747	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ([(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇))
54	50, 53	anbi12d 631	. . . . . 6 ⊢ (𝑞 = 𝑄 → ((([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)))
55	47, 54	rspc2ev 3592	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵 ∧ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
56	40, 55	mp3an3 1450	. . . 4 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
57	56	3adant1 1130	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
58		ecexg 8652	. . . . . 6 ⊢ (𝑇 ∈ 𝑍 → [(𝑃 + 𝑄)]𝑇 ∈ V)
59	3, 58	syl 17	. . . . 5 ⊢ (𝜑 → [(𝑃 + 𝑄)]𝑇 ∈ V)
60	59	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [(𝑃 + 𝑄)]𝑇 ∈ V)
61		simp1 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → 𝜑)
62	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eroveu 8751	. . . . 5 ⊢ ((𝜑 ∧ ([𝑃]𝑅 ∈ 𝐽 ∧ [𝑄]𝑆 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
63	61, 27, 32, 62	syl12anc 835	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
64		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → 𝑧 = [(𝑃 + 𝑄)]𝑇)
65	64	eqeq1d 2738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
66	65	anbi2d 629	. . . . 5 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
67	66	2rexbidv 3213	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
68	60, 63, 67	iota2d 6484	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇))
69	57, 68	mpbid 231	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇)
70	35, 69	eqtrd 2776	1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃!weu 2566  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910   class class class wbr 5105   × cxp 5631  ℩cio 6446  ⟶wf 6492  (class class class)co 7357  {coprab 7358   ∈ cmpo 7359   Er wer 8645  [cec 8646   / cqs 8647

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-ec 8650  df-qs 8654

This theorem is referenced by:  erov2  8755