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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 (φ𝑇 𝑍)
eropr.4 (φ𝑅 Er 𝑈)
eropr.5 (φ𝑆 Er 𝑉)
eropr.6 (φ𝑇 Er 𝑊)
eropr.7 (φA𝑈)
eropr.8 (φB𝑉)
eropr.9 (φ𝐶𝑊)
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 (φ𝑅 𝑋)
eropr.14 (φ𝑆 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
StepHypRef Expression
1 eropr.3 . . . . . . . . . . . 12 (φ𝑇 𝑍)
21ad2antrr 457 . . . . . . . . . . 11 (((φ (x 𝐽 y 𝐾)) (𝑝 A 𝑞 B)) → 𝑇 𝑍)
3 eropr.10 . . . . . . . . . . . . 13 (φ+ :(A × B)⟶𝐶)
43adantr 261 . . . . . . . . . . . 12 ((φ (x 𝐽 y 𝐾)) → + :(A × B)⟶𝐶)
54fovrnda 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))
72, 5, 6syl2anc 391 . . . . . . . . . 10 (((φ (x 𝐽 y 𝐾)) (𝑝 A 𝑞 B)) → [(𝑝 + 𝑞)]𝑇 (𝐶 / 𝑇))
8 eropr.15 . . . . . . . . . 10 𝐿 = (𝐶 / 𝑇)
97, 8syl6eleqr 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)]𝑇z 𝐿))
119, 10syl 14 . . . . . . . 8 (((φ (x 𝐽 y 𝐾)) (𝑝 A 𝑞 B)) → (z = [(𝑝 + 𝑞)]𝑇z 𝐿))
1211adantld 263 . . . . . . 7 (((φ (x 𝐽 y 𝐾)) (𝑝 A 𝑞 B)) → (((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → z 𝐿))
1312rexlimdvva 2434 . . . . . 6 ((φ (x 𝐽 y 𝐾)) → (𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → z 𝐿))
1413abssdv 3008 . . . . 5 ((φ (x 𝐽 y 𝐾)) → {z𝑝 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 (φ𝑅 Er 𝑈)
18 eropr.5 . . . . . . 7 (φ𝑆 Er 𝑉)
19 eropr.6 . . . . . . 7 (φ𝑇 Er 𝑊)
20 eropr.7 . . . . . . 7 (φA𝑈)
21 eropr.8 . . . . . . 7 (φB𝑉)
22 eropr.9 . . . . . . 7 (φ𝐶𝑊)
23 eropr.11 . . . . . . 7 ((φ ((𝑟 A 𝑠 A) (𝑡 B u B))) → ((𝑟𝑅𝑠 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))
2415, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23eroveu 6133 . . . . . 6 ((φ (x 𝐽 y 𝐾)) → ∃!z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))
25 iotacl 4833 . . . . . 6 (∃!z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) {z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)})
2624, 25syl 14 . . . . 5 ((φ (x 𝐽 y 𝐾)) → (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) {z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)})
2714, 26sseldd 2940 . . . 4 ((φ (x 𝐽 y 𝐾)) → (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) 𝐿)
2827ralrimivva 2395 . . 3 (φx 𝐽 y 𝐾 (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) 𝐿)
29 eqid 2037 . . . 4 (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))) = (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))
3029fmpt2 5769 . . 3 (x 𝐽 y 𝐾 (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) 𝐿 ↔ (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
3128, 30sylib 127 . 2 (φ → (x 𝐽, y 𝐾 ↦ (℩z𝑝 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)}
3315, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32erovlem 6134 . . 3 (φ = (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))))
3433feq1d 4977 . 2 (φ → ( :(𝐽 × 𝐾)⟶𝐿 ↔ (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿))
3531, 34mpbird 156 1 (φ :(𝐽 × 𝐾)⟶𝐿)
Colors of variables: wff set class
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
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