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	⊢ (((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 11112	. 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2		opex 5457	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	ecid 8759	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩
4		opex 5457	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
5	4	ecid 8759	. . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩
6	3, 5	oveq12i 7405	. 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7		opex 5457	. . 3 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
8	7	ecid 8759	. 2 ⊢ [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
9	1, 6, 8	3eqtr4g 2796	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 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