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Theorem erov2 8828
Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
eropr2.1 ð― = (ðī / ∞ )
eropr2.2 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðī ((ð‘Ĩ = [𝑝] ∞ ∧ ð‘Ķ = [𝑞] ∞ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∞ )}
eropr2.3 (𝜑 → ∞ ∈ 𝑋)
eropr2.4 (𝜑 → ∞ Er 𝑈)
eropr2.5 (𝜑 → ðī ⊆ 𝑈)
eropr2.6 (𝜑 → + :(ðī × ðī)âŸķðī)
eropr2.7 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðī ∧ ð‘Ē ∈ ðī))) → ((𝑟 ∞ 𝑠 ∧ ð‘Ą ∞ ð‘Ē) → (𝑟 + ð‘Ą) ∞ (𝑠 + ð‘Ē)))
Assertion
Ref Expression
erov2 ((𝜑 ∧ 𝑃 ∈ ðī ∧ 𝑄 ∈ ðī) → ([𝑃] ∞ âĻĢ [𝑄] ∞ ) = [(𝑃 + 𝑄)] ∞ )
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧,ðī   𝑃,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,𝑧   + ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ∞ ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ð―,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝑄,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧
Allowed substitution hints:   âĻĢ (ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑈(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   ð―(ð‘Ē,ð‘Ą,𝑠,𝑟)   𝑋(ð‘Ĩ,ð‘Ķ)

Proof of Theorem erov2
StepHypRef Expression
1 eropr2.1 . 2 ð― = (ðī / ∞ )
2 eropr2.3 . 2 (𝜑 → ∞ ∈ 𝑋)
3 eropr2.4 . 2 (𝜑 → ∞ Er 𝑈)
4 eropr2.5 . 2 (𝜑 → ðī ⊆ 𝑈)
5 eropr2.6 . 2 (𝜑 → + :(ðī × ðī)âŸķðī)
6 eropr2.7 . 2 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðī ∧ ð‘Ē ∈ ðī))) → ((𝑟 ∞ 𝑠 ∧ ð‘Ą ∞ ð‘Ē) → (𝑟 + ð‘Ą) ∞ (𝑠 + ð‘Ē)))
7 eropr2.2 . 2 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðī ((ð‘Ĩ = [𝑝] ∞ ∧ ð‘Ķ = [𝑞] ∞ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∞ )}
81, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2erov 8826 1 ((𝜑 ∧ 𝑃 ∈ ðī ∧ 𝑄 ∈ ðī) → ([𝑃] ∞ âĻĢ [𝑄] ∞ ) = [(𝑃 + 𝑄)] ∞ )
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  âˆƒwrex 3066   ⊆ wss 3945   class class class wbr 5142   × cxp 5670  âŸķwf 6538  (class class class)co 7414  {coprab 7415   Er wer 8715  [cec 8716   / cqs 8717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-er 8718  df-ec 8720  df-qs 8724
This theorem is referenced by: (None)
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