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Theorem addcnsrec 10558
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10557 and mulcnsrec 10559. (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 10550 . 2 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2 opex 5324 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 8349 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 5324 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 8349 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 7151 . 2 ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7 opex 5324 . . 3 ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
87ecid 8349 . 2 [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
91, 6, 83eqtr4g 2861 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 399   = wceq 1538  wcel 2112  cop 4534   E cep 5432  ccnv 5522  (class class class)co 7139  [cec 8274  Rcnr 10280   +R cplr 10284   + caddc 10533
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-eprel 5433  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-ec 8278  df-c 10536  df-add 10541
This theorem is referenced by:  axaddass  10571  axdistr  10573
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