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Theorem addcnsrec 11181
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11180 and mulcnsrec 11182. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcnsrec (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )

Proof of Theorem addcnsrec
StepHypRef Expression
1 addcnsr 11173 . 2 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2 opex 5475 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 8821 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 5475 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 8821 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 7443 . 2 ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7 opex 5475 . . 3 ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
87ecid 8821 . 2 [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
91, 6, 83eqtr4g 2800 1 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   E cep 5588  ccnv 5688  (class class class)co 7431  [cec 8742  Rcnr 10903   +R cplr 10907   + caddc 11156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-ec 8746  df-c 11159  df-add 11164
This theorem is referenced by:  axaddass  11194  axdistr  11196
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