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Theorem addcnsrec 11120
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11119 and mulcnsrec 11121. (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 11112 . 2 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2 opex 5457 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 8759 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 5457 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 8759 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 7405 . 2 ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7 opex 5457 . . 3 ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
87ecid 8759 . 2 [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
91, 6, 83eqtr4g 2796 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 396   = wceq 1541  wcel 2106  cop 4628   E cep 5572  ccnv 5668  (class class class)co 7393  [cec 8684  Rcnr 10842   +R cplr 10846   + caddc 11095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fv 6540  df-ov 7396  df-oprab 7397  df-ec 8688  df-c 11098  df-add 11103
This theorem is referenced by:  axaddass  11133  axdistr  11135
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