Theorem nvdi 24163

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 2454	. . 3 |- ( 1st ` U ) = ( 1st ` U )
2	1	nvvc 24146	. 2 |- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
3		nvdi.2	. . . 4 |- G = ( +v ` U )
4	3	vafval 24134	. . 3 |- G = ( 1st ` ( 1st ` U ) )
5		nvdi.4	. . . 4 |- S = ( .sOLD ` U )
6	5	smfval 24136	. . 3 |- S = ( 2nd ` ( 1st ` U ) )
7		nvdi.1	. . . 4 |- X = ( BaseSet ` U )
8	7, 3	bafval 24135	. . 3 |- X = ran G
9	4, 6, 8	vcdi 24083	. 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 471	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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   ` cfv 5527  (class class class)co 6201   1stc1st 6686   CCcc 9392   CVecOLDcvc 24076   NrmCVeccnv 24115   +vcpv 24116   BaseSetcba 24117   .sOLDcns 24118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-1st 6688  df-2nd 6689  df-vc 24077  df-nv 24123  df-va 24126  df-ba 24127  df-sm 24128  df-0v 24129  df-nmcv 24131

This theorem is referenced by:  nvmdi  24183  nvaddsub4  24194  nvnncan  24196  nvdif  24206  nvpi  24207  ipdirilem  24382  hldi  24461