Theorem eroprf 6135

Description: Functionality 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)
eropr.15	⊢ 𝐿 = (𝐶 / 𝑇)

Assertion

Ref	Expression
eroprf	⊢ (φ → ⨣ :(𝐽 × 𝐾)⟶𝐿)

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,u,x,y,z,A   B,𝑝,𝑞,𝑟,𝑠,𝑡,u,x,y,z   𝐿,𝑝,𝑞,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   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)   𝐿(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 eroprf

Step	Hyp	Ref	Expression
1		eropr.3	. . . . . . . . . . . 12 ⊢ (φ → 𝑇 ∈ 𝑍)
2	1	ad2antrr 457	. . . . . . . . . . 11 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → 𝑇 ∈ 𝑍)
3		eropr.10	. . . . . . . . . . . . 13 ⊢ (φ → + :(A × B)⟶𝐶)
4	3	adantr 261	. . . . . . . . . . . 12 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → + :(A × B)⟶𝐶)
5	4	fovrnda 5586	. . . . . . . . . . 11 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → (𝑝 + 𝑞) ∈ 𝐶)
6		ecelqsg 6095	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ 𝑍 ∧ (𝑝 + 𝑞) ∈ 𝐶) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
7	2, 5, 6	syl2anc 391	. . . . . . . . . 10 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
8		eropr.15	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶 / 𝑇)
9	7, 8	syl6eleqr 2128	. . . . . . . . 9 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → [(𝑝 + 𝑞)]𝑇 ∈ 𝐿)
10		eleq1a 2106	. . . . . . . . 9 ⊢ ([(𝑝 + 𝑞)]𝑇 ∈ 𝐿 → (z = [(𝑝 + 𝑞)]𝑇 → z ∈ 𝐿))
11	9, 10	syl 14	. . . . . . . 8 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → (z = [(𝑝 + 𝑞)]𝑇 → z ∈ 𝐿))
12	11	adantld 263	. . . . . . 7 ⊢ (((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) ∧ (𝑝 ∈ A ∧ 𝑞 ∈ B)) → (((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → z ∈ 𝐿))
13	12	rexlimdvva 2434	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → (∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇) → z ∈ 𝐿))
14	13	abssdv 3008	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → {z ∣ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)} ⊆ 𝐿)
15		eropr.1	. . . . . . 7 ⊢ 𝐽 = (A / 𝑅)
16		eropr.2	. . . . . . 7 ⊢ 𝐾 = (B / 𝑆)
17		eropr.4	. . . . . . 7 ⊢ (φ → 𝑅 Er 𝑈)
18		eropr.5	. . . . . . 7 ⊢ (φ → 𝑆 Er 𝑉)
19		eropr.6	. . . . . . 7 ⊢ (φ → 𝑇 Er 𝑊)
20		eropr.7	. . . . . . 7 ⊢ (φ → A ⊆ 𝑈)
21		eropr.8	. . . . . . 7 ⊢ (φ → B ⊆ 𝑉)
22		eropr.9	. . . . . . 7 ⊢ (φ → 𝐶 ⊆ 𝑊)
23		eropr.11	. . . . . . 7 ⊢ ((φ ∧ ((𝑟 ∈ A ∧ 𝑠 ∈ A) ∧ (𝑡 ∈ B ∧ u ∈ B))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))
24	15, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23	eroveu 6133	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → ∃!z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))
25		iotacl 4833	. . . . . 6 ⊢ (∃!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 ∣ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)})
26	24, 25	syl 14	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → (℩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)})
27	14, 26	sseldd 2940	. . . 4 ⊢ ((φ ∧ (x ∈ 𝐽 ∧ y ∈ 𝐾)) → (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
28	27	ralrimivva 2395	. . 3 ⊢ (φ → ∀x ∈ 𝐽 ∀y ∈ 𝐾 (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
29		eqid 2037	. . . 4 ⊢ (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))) = (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)))
30	29	fmpt2 5769	. . 3 ⊢ (∀x ∈ 𝐽 ∀y ∈ 𝐾 (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿 ↔ (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
31	28, 30	sylib 127	. 2 ⊢ (φ → (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
32		eropr.12	. . . 4 ⊢ ⨣ = {⟨⟨x, y⟩, z⟩ ∣ ∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇)}
33	15, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32	erovlem 6134	. . 3 ⊢ (φ → ⨣ = (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))))
34	33	feq1d 4977	. 2 ⊢ (φ → ( ⨣ :(𝐽 × 𝐾)⟶𝐿 ↔ (x ∈ 𝐽, y ∈ 𝐾 ↦ (℩z∃𝑝 ∈ A ∃𝑞 ∈ B ((x = [𝑝]𝑅 ∧ y = [𝑞]𝑆) ∧ z = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿))
35	31, 34	mpbird 156	1 ⊢ (φ → ⨣ :(𝐽 × 𝐾)⟶𝐿)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  {cab 2023  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911   class class class wbr 3755   × cxp 4286  ℩cio 4808  ⟶wf 4841  (class class class)co 5455  {coprab 5456   ↦ cmpt2 5457   Er wer 6039  [cec 6040   / cqs 6041

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 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

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-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-er 6042  df-ec 6044  df-qs 6048

This theorem is referenced by:  eroprf2  6136