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Theorem nvmdi 22133
Description: Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmdi.1  |-  X  =  ( BaseSet `  U )
nvmdi.3  |-  M  =  ( -v `  U
)
nvmdi.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nvmdi  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B M C ) )  =  ( ( A S B ) M ( A S C ) ) )

Proof of Theorem nvmdi
StepHypRef Expression
1 simpr1 964 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  A  e.  CC )
2 simpr2 965 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  B  e.  X )
3 neg1cn 10069 . . . . . . 7  |-  -u 1  e.  CC
4 nvmdi.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 nvmdi.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
64, 5nvscl 22109 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
73, 6mp3an2 1268 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
873ad2antr3 1125 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( -u 1 S C )  e.  X )
91, 2, 83jca 1135 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A  e.  CC  /\  B  e.  X  /\  ( -u 1 S C )  e.  X ) )
10 eqid 2438 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
114, 10, 5nvdi 22113 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  ( -u 1 S C )  e.  X ) )  ->  ( A S ( B ( +v
`  U ) (
-u 1 S C ) ) )  =  ( ( A S B ) ( +v
`  U ) ( A S ( -u
1 S C ) ) ) )
129, 11syldan 458 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B ( +v `  U ) ( -u
1 S C ) ) )  =  ( ( A S B ) ( +v `  U ) ( A S ( -u 1 S C ) ) ) )
134, 5nvscom 22112 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  -u 1  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
143, 13mp3anr2 1278 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
15143adantr2 1118 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
1615oveq2d 6099 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( ( A S B ) ( +v
`  U ) ( A S ( -u
1 S C ) ) )  =  ( ( A S B ) ( +v `  U ) ( -u
1 S ( A S C ) ) ) )
1712, 16eqtrd 2470 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B ( +v `  U ) ( -u
1 S C ) ) )  =  ( ( A S B ) ( +v `  U ) ( -u
1 S ( A S C ) ) ) )
18 nvmdi.3 . . . . 5  |-  M  =  ( -v `  U
)
194, 10, 5, 18nvmval 22125 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  =  ( B ( +v
`  U ) (
-u 1 S C ) ) )
20193adant3r1 1163 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( B M C )  =  ( B ( +v `  U
) ( -u 1 S C ) ) )
2120oveq2d 6099 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B M C ) )  =  ( A S ( B ( +v `  U ) ( -u 1 S C ) ) ) )
22 simpl 445 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  U  e.  NrmCVec )
234, 5nvscl 22109 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
24233adant3r3 1165 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S B )  e.  X )
254, 5nvscl 22109 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  C  e.  X )  ->  ( A S C )  e.  X )
26253adant3r2 1164 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S C )  e.  X )
274, 10, 5, 18nvmval 22125 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A S B )  e.  X  /\  ( A S C )  e.  X )  ->  (
( A S B ) M ( A S C ) )  =  ( ( A S B ) ( +v `  U ) ( -u 1 S ( A S C ) ) ) )
2822, 24, 26, 27syl3anc 1185 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( ( A S B ) M ( A S C ) )  =  ( ( A S B ) ( +v `  U
) ( -u 1 S ( A S C ) ) ) )
2917, 21, 283eqtr4d 2480 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B M C ) )  =  ( ( A S B ) M ( A S C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   CCcc 8990   1c1 8993   -ucneg 9294   NrmCVeccnv 22065   +vcpv 22066   BaseSetcba 22067   .s
OLDcns 22068   -vcnsb 22070
This theorem is referenced by:  smcnlem  22195  minvecolem2  22379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127  df-sub 9295  df-neg 9296  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081
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