Theorem nvdi 25187

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 2460	. . 3 |- ( 1st ` U ) = ( 1st ` U )
2	1	nvvc 25170	. 2 |- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
3		nvdi.2	. . . 4 |- G = ( +v ` U )
4	3	vafval 25158	. . 3 |- G = ( 1st ` ( 1st ` U ) )
5		nvdi.4	. . . 4 |- S = ( .sOLD ` U )
6	5	smfval 25160	. . 3 |- S = ( 2nd ` ( 1st ` U ) )
7		nvdi.1	. . . 4 |- X = ( BaseSet ` U )
8	7, 3	bafval 25159	. . 3 |- X = ran G
9	4, 6, 8	vcdi 25107	. 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 968   = wceq 1374   e. wcel 1762   ` cfv 5579  (class class class)co 6275   1stc1st 6772   CCcc 9479   CVecOLDcvc 25100   NrmCVeccnv 25139   +vcpv 25140   BaseSetcba 25141   .sOLDcns 25142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-1st 6774  df-2nd 6775  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-nmcv 25155

This theorem is referenced by:  nvmdi  25207  nvaddsub4  25218  nvnncan  25220  nvdif  25230  nvpi  25231  ipdirilem  25406  hldi  25485