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Theorem nvmdi 24045
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  =  ( .sOLD `  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 994 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  A  e.  CC )
2 simpr2 995 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  B  e.  X )
3 neg1cn 10440 . . . . . . 7  |-  -u 1  e.  CC
4 nvmdi.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 nvmdi.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
64, 5nvscl 24021 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
73, 6mp3an2 1302 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
873ad2antr3 1155 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( -u 1 S C )  e.  X )
91, 2, 83jca 1168 . . . 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 2443 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
114, 10, 5nvdi 24025 . . . 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 470 . . 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 24024 . . . . . 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 1312 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
15143adantr2 1148 . . . 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 6122 . . 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 2475 . 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 24037 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  =  ( B ( +v
`  U ) (
-u 1 S C ) ) )
20193adant3r1 1196 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( B M C )  =  ( B ( +v `  U
) ( -u 1 S C ) ) )
2120oveq2d 6122 . 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 457 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  U  e.  NrmCVec )
234, 5nvscl 24021 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
24233adant3r3 1198 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S B )  e.  X )
254, 5nvscl 24021 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  C  e.  X )  ->  ( A S C )  e.  X )
26253adant3r2 1197 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S C )  e.  X )
274, 10, 5, 18nvmval 24037 . . 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 1218 . 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 2485 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5433  (class class class)co 6106   CCcc 9295   1c1 9298   -ucneg 9611   NrmCVeccnv 23977   +vcpv 23978   BaseSetcba 23979   .sOLDcns 23980   -vcnsb 23982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-ltxr 9438  df-sub 9612  df-neg 9613  df-grpo 23693  df-gid 23694  df-ginv 23695  df-gdiv 23696  df-ablo 23784  df-vc 23939  df-nv 23985  df-va 23988  df-ba 23989  df-sm 23990  df-0v 23991  df-vs 23992  df-nmcv 23993
This theorem is referenced by:  smcnlem  24107  minvecolem2  24291
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