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Theorem vcsubdir 26020
Description: Subtractive distributive law for the scalar product of a complex vector space. (Contributed by NM, 31-Jul-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
vci.1  |-  G  =  ( 1st `  W
)
vci.2  |-  S  =  ( 2nd `  W
)
vci.3  |-  X  =  ran  G
Assertion
Ref Expression
vcsubdir  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  -  B ) S C )  =  ( ( A S C ) G ( -u
1 S ( B S C ) ) ) )

Proof of Theorem vcsubdir
StepHypRef Expression
1 negcl 9874 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 vci.1 . . . . 5  |-  G  =  ( 1st `  W
)
3 vci.2 . . . . 5  |-  S  =  ( 2nd `  W
)
4 vci.3 . . . . 5  |-  X  =  ran  G
52, 3, 4vcdir 26017 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  -u B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  -u B ) S C )  =  ( ( A S C ) G (
-u B S C ) ) )
61, 5syl3anr2 1317 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  -u B ) S C )  =  ( ( A S C ) G (
-u B S C ) ) )
7 negsub 9921 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
873adant3 1025 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  ->  ( A  +  -u B )  =  ( A  -  B ) )
98adantl 467 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( A  +  -u B )  =  ( A  -  B
) )
109oveq1d 6320 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  -u B ) S C )  =  ( ( A  -  B ) S C ) )
116, 10eqtr3d 2472 . 2  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A S C ) G ( -u B S C ) )  =  ( ( A  -  B ) S C ) )
12 mulm1 10059 . . . . . . 7  |-  ( B  e.  CC  ->  ( -u 1  x.  B )  =  -u B )
1312ad2antrl 732 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  ( B  e.  CC  /\  C  e.  X ) )  ->  ( -u 1  x.  B )  =  -u B )
1413oveq1d 6320 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  ( B  e.  CC  /\  C  e.  X ) )  ->  ( ( -u 1  x.  B ) S C )  =  ( -u B S C ) )
15 neg1cn 10713 . . . . . 6  |-  -u 1  e.  CC
162, 3, 4vcass 26018 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  ( -u 1  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( -u 1  x.  B ) S C )  =  ( -u
1 S ( B S C ) ) )
1715, 16mp3anr1 1357 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  ( B  e.  CC  /\  C  e.  X ) )  ->  ( ( -u 1  x.  B ) S C )  =  ( -u 1 S ( B S C ) ) )
1814, 17eqtr3d 2472 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( B  e.  CC  /\  C  e.  X ) )  ->  ( -u B S C )  =  (
-u 1 S ( B S C ) ) )
19183adantr1 1164 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( -u B S C )  =  (
-u 1 S ( B S C ) ) )
2019oveq2d 6321 . 2  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A S C ) G ( -u B S C ) )  =  ( ( A S C ) G (
-u 1 S ( B S C ) ) ) )
2111, 20eqtr3d 2472 1  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  -  B ) S C )  =  ( ( A S C ) G ( -u
1 S ( B S C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   ran crn 4855   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   CCcc 9536   1c1 9539    + caddc 9541    x. cmul 9543    - cmin 9859   -ucneg 9860   CVecOLDcvc 26009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-vc 26010
This theorem is referenced by: (None)
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