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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 (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)}
Assertion
Ref Expression
erovlem (φ = (x 𝐽, y 𝐾 ↦ (℩z𝑝 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.
StepHypRef Expression
1 simpl 102 . . . . . . . 8 (((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → (x = [𝑝]𝑅 y = [𝑞]𝑆))
21reximi 2410 . . . . . . 7 (𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → 𝑞 B (x = [𝑝]𝑅 y = [𝑞]𝑆))
32reximi 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)
54eleq2i 2101 . . . . . . . . 9 (x 𝐽x (A / 𝑅))
6 vex 2554 . . . . . . . . . 10 x V
76elqs 6086 . . . . . . . . 9 (x (A / 𝑅) ↔ 𝑝 A x = [𝑝]𝑅)
85, 7bitri 173 . . . . . . . 8 (x 𝐽𝑝 A x = [𝑝]𝑅)
9 eropr.2 . . . . . . . . . 10 𝐾 = (B / 𝑆)
109eleq2i 2101 . . . . . . . . 9 (y 𝐾y (B / 𝑆))
11 vex 2554 . . . . . . . . . 10 y V
1211elqs 6086 . . . . . . . . 9 (y (B / 𝑆) ↔ 𝑞 B y = [𝑞]𝑆)
1310, 12bitri 173 . . . . . . . 8 (y 𝐾𝑞 B y = [𝑞]𝑆)
148, 13anbi12i 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))
1614, 15bitr4i 176 . . . . . 6 ((x 𝐽 y 𝐾) ↔ 𝑝 A 𝑞 B (x = [𝑝]𝑅 y = [𝑞]𝑆))
173, 16sylibr 137 . . . . 5 (𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) → (x 𝐽 y 𝐾))
1817pm4.71ri 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 (φ𝑇 𝑍)
20 eropr.4 . . . . . . . 8 (φ𝑅 Er 𝑈)
21 eropr.5 . . . . . . . 8 (φ𝑆 Er 𝑉)
22 eropr.6 . . . . . . . 8 (φ𝑇 Er 𝑊)
23 eropr.7 . . . . . . . 8 (φA𝑈)
24 eropr.8 . . . . . . . 8 (φB𝑉)
25 eropr.9 . . . . . . . 8 (φ𝐶𝑊)
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)))
284, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27eroveu 6126 . . . . . . 7 ((φ (x 𝐽 y 𝐾)) → ∃!z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))
29 iota1 4823 . . . . . . 7 (∃!z𝑝 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) ↔ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) = z))
3028, 29syl 14 . . . . . 6 ((φ (x 𝐽 y 𝐾)) → (𝑝 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)) = z))
31 eqcom 2039 . . . . . 6 ((℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) = zz = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))
3230, 31syl6bb 185 . . . . 5 ((φ (x 𝐽 y 𝐾)) → (𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) ↔ z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))))
3332pm5.32da 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 = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))))
3418, 33syl5bb 181 . . 3 (φ → (𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇) ↔ ((x 𝐽 y 𝐾) z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))))
3534oprabbidv 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 = (℩z𝑝 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 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨x, y⟩, w⟩ ∣ ((x 𝐽 y 𝐾) w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))}
38 nfv 1418 . . . 4 w((x 𝐽 y 𝐾) z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))
39 nfv 1418 . . . . 5 z(x 𝐽 y 𝐾)
40 nfiota1 4811 . . . . . 6 z(℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))
4140nfeq2 2186 . . . . 5 z w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))
4239, 41nfan 1454 . . . 4 z((x 𝐽 y 𝐾) w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))
43 eqeq1 2043 . . . . 5 (z = w → (z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)) ↔ w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))))
4443anbi2d 437 . . . 4 (z = w → (((x 𝐽 y 𝐾) z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))) ↔ ((x 𝐽 y 𝐾) w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))))
4538, 42, 44cbvoprab3 5519 . . 3 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝐽 y 𝐾) z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))} = {⟨⟨x, y⟩, w⟩ ∣ ((x 𝐽 y 𝐾) w = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))}
4637, 45eqtr4i 2060 . 2 (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝐽 y 𝐾) z = (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)))}
4735, 36, 463eqtr4g 2094 1 (φ = (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))))
Colors of variables: wff set class
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
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