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Theorem erov2 7831
Description: The value 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
erov2 ((𝜑𝑃𝐴𝑄𝐴) → ([𝑃] [𝑄] ) = [(𝑃 + 𝑄)] )
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝑃,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   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   𝑄,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 erov2
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, 2erov 7829 1 ((𝜑𝑃𝐴𝑄𝐴) → ([𝑃] [𝑄] ) = [(𝑃 + 𝑄)] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wrex 2910  wss 3567   class class class wbr 4644   × cxp 5102  wf 5872  (class class class)co 6635  {coprab 6636   Er wer 7724  [cec 7725   / cqs 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-er 7727  df-ec 7729  df-qs 7733
This theorem is referenced by: (None)
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