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Theorem eroprf2 7807
Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
eropr2.1 𝐽 = (𝐴 / )
eropr2.2 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐴 ((𝑥 = [𝑝] 𝑦 = [𝑞] ) ∧ 𝑧 = [(𝑝 + 𝑞)] )}
eropr2.3 (𝜑𝑋)
eropr2.4 (𝜑 Er 𝑈)
eropr2.5 (𝜑𝐴𝑈)
eropr2.6 (𝜑+ :(𝐴 × 𝐴)⟶𝐴)
eropr2.7 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐴𝑢𝐴))) → ((𝑟 𝑠𝑡 𝑢) → (𝑟 + 𝑡) (𝑠 + 𝑢)))
Assertion
Ref Expression
eroprf2 (𝜑 :(𝐽 × 𝐽)⟶𝐽)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝑋(𝑥,𝑦)

Proof of Theorem eroprf2
StepHypRef Expression
1 eropr2.1 . 2 𝐽 = (𝐴 / )
2 eropr2.3 . 2 (𝜑𝑋)
3 eropr2.4 . 2 (𝜑 Er 𝑈)
4 eropr2.5 . 2 (𝜑𝐴𝑈)
5 eropr2.6 . 2 (𝜑+ :(𝐴 × 𝐴)⟶𝐴)
6 eropr2.7 . 2 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐴𝑢𝐴))) → ((𝑟 𝑠𝑡 𝑢) → (𝑟 + 𝑡) (𝑠 + 𝑢)))
7 eropr2.2 . 2 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐴 ((𝑥 = [𝑝] 𝑦 = [𝑞] ) ∧ 𝑧 = [(𝑝 + 𝑞)] )}
81, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2, 1eroprf 7805 1 (𝜑 :(𝐽 × 𝐽)⟶𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560   class class class wbr 4623   × cxp 5082  wf 5853  (class class class)co 6615  {coprab 6616   Er wer 7699  [cec 7700   / cqs 7701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-ec 7704  df-qs 7708
This theorem is referenced by: (None)
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