Theorem addcnsrec 9964

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 9963 and mulcnsrec 9965. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addcnsrec	|- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( [ <. A , B >. ] `' _E + [ <. C , D >. ] `' _E ) = [ <. ( A +R C ) , ( B +R D ) >. ] `' _E )

Proof of Theorem addcnsrec

Step	Hyp	Ref	Expression
1		addcnsr 9956	. 2 |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. )
2		opex 4932	. . . 4 |- <. A , B >. e. _V
3	2	ecid 7812	. . 3 |- [ <. A , B >. ] `' _E = <. A , B >.
4		opex 4932	. . . 4 |- <. C , D >. e. _V
5	4	ecid 7812	. . 3 |- [ <. C , D >. ] `' _E = <. C , D >.
6	3, 5	oveq12i 6662	. 2 |- ( [ <. A , B >. ] `' _E + [ <. C , D >. ] `' _E ) = ( <. A , B >. + <. C , D >. )
7		opex 4932	. . 3 |- <. ( A +R C ) , ( B +R D ) >. e. _V
8	7	ecid 7812	. 2 |- [ <. ( A +R C ) , ( B +R D ) >. ] `' _E = <. ( A +R C ) , ( B +R D ) >.
9	1, 6, 8	3eqtr4g 2681	1 |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( [ <. A , B >. ] `' _E + [ <. C , D >. ] `' _E ) = [ <. ( A +R C ) , ( B +R D ) >. ] `' _E )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   _E cep 5028   `'ccnv 5113  (class class class)co 6650   [cec 7740   R.cnr 9687   +R cplr 9691   + caddc 9939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-ec 7744  df-c 9942  df-add 9947

This theorem is referenced by:  axaddass  9977  axdistr  9979