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Theorem addcnsrec 10002
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10001 and mulcnsrec 10003. (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 9994 . 2 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2 opex 4962 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 7855 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 4962 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 7855 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 6702 . 2 ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7 opex 4962 . . 3 ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
87ecid 7855 . 2 [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
91, 6, 83eqtr4g 2710 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 383   = wceq 1523  wcel 2030  cop 4216   E cep 5057  ccnv 5142  (class class class)co 6690  [cec 7785  Rcnr 9725   +R cplr 9729   + caddc 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-ec 7789  df-c 9980  df-add 9985
This theorem is referenced by:  axaddass  10015  axdistr  10017
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