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Theorem vcsubdir 26112
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 9821 . . . 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 26109 . . . 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 9868 . . . . . 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 6259 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  -u B ) S C )  =  ( ( A  -  B ) S C ) )
116, 10eqtr3d 2459 . 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 10006 . . . . . . 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 6259 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  ( B  e.  CC  /\  C  e.  X ) )  ->  ( ( -u 1  x.  B ) S C )  =  ( -u B S C ) )
15 neg1cn 10659 . . . . . 6  |-  -u 1  e.  CC
162, 3, 4vcass 26110 . . . . . 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 2459 . . . 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 6260 . 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 2459 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 1872   ran crn 4792   ` cfv 5539  (class class class)co 6244   1stc1st 6744   2ndc2nd 6745   CCcc 9483   1c1 9486    + caddc 9488    x. cmul 9490    - cmin 9806   -ucneg 9807   CVecOLDcvc 26101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561
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 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-po 4712  df-so 4713  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-1st 6746  df-2nd 6747  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-pnf 9623  df-mnf 9624  df-ltxr 9626  df-sub 9808  df-neg 9809  df-vc 26102
This theorem is referenced by: (None)
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