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Theorem eroprf2 6683
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 6682 1 (𝜑 :(𝐽 × 𝐽)⟶𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  wrex 2473  wss 3153   class class class wbr 4029   × cxp 4657  wf 5250  (class class class)co 5918  {coprab 5919   Er wer 6584  [cec 6585   / cqs 6586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-er 6587  df-ec 6589  df-qs 6593
This theorem is referenced by: (None)
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