Theorem addcnsrec 11128

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11127 and mulcnsrec 11129. (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 11120	. 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
2		opex 5446	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	ecid 8778	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩
4		opex 5446	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
5	4	ecid 8778	. . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩
6	3, 5	oveq12i 7423	. 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)
7		opex 5446	. . 3 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
8	7	ecid 8778	. 2 ⊢ [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩
9	1, 6, 8	3eqtr4g 2829	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 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4598   E cep 5561  ^◡ccnv 5661  (class class class)co 7411  [cec 8692  Rcnr 10850   +_R cplr 10854   + caddc 11103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-ec 8696  df-c 11106  df-add 11111

This theorem is referenced by:  axaddass  11141  axdistr  11143