Theorem nvdi 25819

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 2402	. . 3 |- ( 1st ` U ) = ( 1st ` U )
2	1	nvvc 25802	. 2 |- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
3		nvdi.2	. . . 4 |- G = ( +v ` U )
4	3	vafval 25790	. . 3 |- G = ( 1st ` ( 1st ` U ) )
5		nvdi.4	. . . 4 |- S = ( .sOLD ` U )
6	5	smfval 25792	. . 3 |- S = ( 2nd ` ( 1st ` U ) )
7		nvdi.1	. . . 4 |- X = ( BaseSet ` U )
8	7, 3	bafval 25791	. . 3 |- X = ran G
9	4, 6, 8	vcdi 25739	. 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 469	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   ` cfv 5525  (class class class)co 6234   1stc1st 6736   CCcc 9440   CVecOLDcvc 25732   NrmCVeccnv 25771   +vcpv 25772   BaseSetcba 25773   .sOLDcns 25774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-1st 6738  df-2nd 6739  df-vc 25733  df-nv 25779  df-va 25782  df-ba 25783  df-sm 25784  df-0v 25785  df-nmcv 25787

This theorem is referenced by:  nvmdi  25839  nvaddsub4  25850  nvnncan  25852  nvdif  25862  nvpi  25863  ipdirilem  26038  hldi  26117