Theorem erovlem 6109

Description: Lemma for eroprf 6110. (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)}

Assertion

Ref	Expression
erovlem	⊢ (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,x,y,z,A   B,𝑝,𝑞,𝑟,𝑠,𝑡,u,x,y,z   𝐽,𝑝,𝑞,x,y,z   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,u,x,y,z   𝐾,𝑝,𝑞,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

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)   𝑍(x,y,z,u,𝑡,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem erovlem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 102	. . . . . . . 8 ⊢ (((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → (x = [𝑝]𝑅 ∧ y = [𝑞]𝑆))
2	1	reximi 2394	. . . . . . 7 ⊢ (∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ B (x = [𝑝]𝑅 ∧ y = [𝑞]𝑆))
3	2	reximi 2394	. . . . . 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))
4		eropr.1	. . . . . . . . . 10 ⊢ 𝐽 = (A / 𝑅)
5	4	eleq2i 2086	. . . . . . . . 9 ⊢ (x ∈ 𝐽 ↔ x ∈ (A / 𝑅))
6		vex 2538	. . . . . . . . . 10 ⊢ x ∈ V
7	6	elqs 6068	. . . . . . . . 9 ⊢ (x ∈ (A / 𝑅) ↔ ∃𝑝 ∈ A x = [𝑝]𝑅)
8	5, 7	bitri 173	. . . . . . . 8 ⊢ (x ∈ 𝐽 ↔ ∃𝑝 ∈ A x = [𝑝]𝑅)
9		eropr.2	. . . . . . . . . 10 ⊢ 𝐾 = (B / 𝑆)
10	9	eleq2i 2086	. . . . . . . . 9 ⊢ (y ∈ 𝐾 ↔ y ∈ (B / 𝑆))
11		vex 2538	. . . . . . . . . 10 ⊢ y ∈ V
12	11	elqs 6068	. . . . . . . . 9 ⊢ (y ∈ (B / 𝑆) ↔ ∃𝑞 ∈ B y = [𝑞]𝑆)
13	10, 12	bitri 173	. . . . . . . 8 ⊢ (y ∈ 𝐾 ↔ ∃𝑞 ∈ B y = [𝑞]𝑆)
14	8, 13	anbi12i 436	. . . . . . 7 ⊢ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ↔ (∃𝑝 ∈ A x = [𝑝]𝑅 ∧ ∃𝑞 ∈ B y = [𝑞]𝑆))
15		reeanv 2457	. . . . . . 7 ⊢ (∃𝑝 ∈ A ∃𝑞 ∈ B (x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ↔ (∃𝑝 ∈ A x = [𝑝]𝑅 ∧ ∃𝑞 ∈ B y = [𝑞]𝑆))
16	14, 15	bitr4i 176	. . . . . 6 ⊢ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ↔ ∃𝑝 ∈ A ∃𝑞 ∈ B (x = [𝑝]𝑅 ∧ y = [𝑞]𝑆))
17	3, 16	sylibr 137	. . . . 5 ⊢ (∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → (x ∈ 𝐽 ∧ y ∈ 𝐾))
18	17	pm4.71ri 372	. . . 4 ⊢ (∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) ↔ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
19		eropr.3	. . . . . . . 8 ⊢ (φ → 𝑇 ∈ 𝑍)
20		eropr.4	. . . . . . . 8 ⊢ (φ → 𝑅 Er 𝑈)
21		eropr.5	. . . . . . . 8 ⊢ (φ → 𝑆 Er 𝑉)
22		eropr.6	. . . . . . . 8 ⊢ (φ → 𝑇 Er 𝑊)
23		eropr.7	. . . . . . . 8 ⊢ (φ → A ⊆ 𝑈)
24		eropr.8	. . . . . . . 8 ⊢ (φ → B ⊆ 𝑉)
25		eropr.9	. . . . . . . 8 ⊢ (φ → 𝐶 ⊆ 𝑊)
26		eropr.10	. . . . . . . 8 ⊢ (φ → + :(A × B)⟶𝐶)
27		eropr.11	. . . . . . . 8 ⊢ ((φ ∧ ((𝑟 ∈ A ∧ 𝑠 ∈ A) ∧ (𝑡 ∈ B ∧ u ∈ B))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))
28	4, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27	eroveu 6108	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → ∃!z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
29		iota1 4808	. . . . . . 7 ⊢ (∃!z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → (∃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))
30	28, 29	syl 14	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → (∃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))
31		eqcom 2024	. . . . . 6 ⊢ ((℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) = z ↔ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
32	30, 31	syl6bb 185	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → (∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) ↔ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))))
33	32	pm5.32da 428	. . . 4 ⊢ (φ → (((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) ↔ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))))
34	18, 33	syl5bb 181	. . 3 ⊢ (φ → (∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) ↔ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))))
35	34	oprabbidv 5482	. 2 ⊢ (φ → {⟨⟨x, y⟩, z⟩ ∣ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))})
36		eropr.12	. 2 ⊢ ⨣ = {⟨⟨x, y⟩, z⟩ ∣ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)}
37		df-mpt2 5441	. . 3 ⊢ (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))}
38		nfv 1402	. . . 4 ⊢ Ⅎw((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
39		nfv 1402	. . . . 5 ⊢ Ⅎz(x ∈ 𝐽 ∧ y ∈ 𝐾)
40		nfiota1 4796	. . . . . 6 ⊢ Ⅎz(℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
41	40	nfeq2 2171	. . . . 5 ⊢ Ⅎz w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
42	39, 41	nfan 1439	. . . 4 ⊢ Ⅎz((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
43		eqeq1 2028	. . . . 5 ⊢ (z = w → (z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) ↔ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))))
44	43	anbi2d 440	. . . 4 ⊢ (z = w → (((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))) ↔ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))))
45	38, 42, 44	cbvoprab3 5503	. . 3 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))} = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))}
46	37, 45	eqtr4i 2045	. 2 ⊢ (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))}
47	35, 36, 46	3eqtr4g 2079	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 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃!weu 1882  ∃wrex 2285   ⊆ wss 2894   class class class wbr 3738   × cxp 4270  ℩cio 4792  ⟶wf 4825  (class class class)co 5436  {coprab 5437   ↦ cmpt2 5438   Er wer 6014  [cec 6015   / cqs 6016

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-er 6017  df-ec 6019  df-qs 6023

This theorem is referenced by:  eroprf  6110