Theorem nvdi 27485

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvdi.1	|- X = ( BaseSet ` U )
nvdi.2	|- G = ( +v ` U )
nvdi.4	|- S = ( .sOLD ` U )

Assertion

Ref	Expression
nvdi	|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A S C ) ) )

Proof of Theorem nvdi

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( 1st ` U ) = ( 1st ` U )
2	1	nvvc 27470	. 2 |- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
3		nvdi.2	. . . 4 |- G = ( +v ` U )
4	3	vafval 27458	. . 3 |- G = ( 1st ` ( 1st ` U ) )
5		nvdi.4	. . . 4 |- S = ( .sOLD ` U )
6	5	smfval 27460	. . 3 |- S = ( 2nd ` ( 1st ` U ) )
7		nvdi.1	. . . 4 |- X = ( BaseSet ` U )
8	7, 3	bafval 27459	. . 3 |- X = ran G
9	4, 6, 8	vcdi 27420	. 2 |- ( ( ( 1st ` U ) e. CVecOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A S C ) ) )
10	2, 9	sylan 488	1 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A S C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   1stc1st 7166   CCcc 9934   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  nvmdi  27503  nvaddsub4  27512  nvdif  27521  nvpi  27522  ipdirilem  27684  hldi  27763