Theorem erovlem 6127

Description: Lemma for eroprf 6128. (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 2410	. . . . . . 7 ⊢ (∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ B (x = [𝑝]𝑅 ∧ y = [𝑞]𝑆))
3	2	reximi 2410	. . . . . 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 2101	. . . . . . . . 9 ⊢ (x ∈ 𝐽 ↔ x ∈ (A / 𝑅))
6		vex 2554	. . . . . . . . . 10 ⊢ x ∈ V
7	6	elqs 6086	. . . . . . . . 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 2101	. . . . . . . . 9 ⊢ (y ∈ 𝐾 ↔ y ∈ (B / 𝑆))
11		vex 2554	. . . . . . . . . 10 ⊢ y ∈ V
12	11	elqs 6086	. . . . . . . . 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 433	. . . . . . 7 ⊢ ((x ∈ 𝐽 ∧ y ∈ 𝐾) ↔ (∃𝑝 ∈ A x = [𝑝]𝑅 ∧ ∃𝑞 ∈ B y = [𝑞]𝑆))
15		reeanv 2473	. . . . . . 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 6126	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → ∃!z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
29		iota1 4823	. . . . . . 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 2039	. . . . . 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 425	. . . 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 5498	. 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 5457	. . 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 1418	. . . 4 ⊢ Ⅎw((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ z = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
39		nfv 1418	. . . . 5 ⊢ Ⅎz(x ∈ 𝐽 ∧ y ∈ 𝐾)
40		nfiota1 4811	. . . . . 6 ⊢ Ⅎz(℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
41	40	nfeq2 2186	. . . . 5 ⊢ Ⅎz w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
42	39, 41	nfan 1454	. . . 4 ⊢ Ⅎz((x ∈ 𝐽 ∧ y ∈ 𝐾) ∧ w = (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
43		eqeq1 2043	. . . . 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 437	. . . 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 5519	. . 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 2060	. 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 2094	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 1242   ∈ wcel 1390  ∃!weu 1897  ∃wrex 2301   ⊆ wss 2911   class class class wbr 3754   × cxp 4285  ℩cio 4807  ⟶wf 4840  (class class class)co 5452  {coprab 5453   ↦ cmpt2 5454   Er wer 6032  [cec 6033   / cqs 6034

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-er 6035  df-ec 6037  df-qs 6041

This theorem is referenced by:  eroprf  6128