Theorem addcnsrec 11166

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11165 and mulcnsrec 11167. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addcnsrec	⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E )

Proof of Theorem addcnsrec

Step	Hyp	Ref	Expression
1		addcnsr 11158	. 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2		opex 5460	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	ecid 8799	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩
4		opex 5460	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
5	4	ecid 8799	. . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩
6	3, 5	oveq12i 7428	. 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7		opex 5460	. . 3 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
8	7	ecid 8799	. 2 ⊢ [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
9	1, 6, 8	3eqtr4g 2790	1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E )

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4630   E cep 5575  ^◡ccnv 5671  (class class class)co 7416  [cec 8721  Rcnr 10888   +_R cplr 10892   + caddc 11141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-eprel 5576  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 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-oprab 7420  df-ec 8725  df-c 11144  df-add 11149

This theorem is referenced by:  axaddass  11179  axdistr  11181