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Theorem nvmdi 25368
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 1002 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  A  e.  CC )
2 simpr2 1003 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  B  e.  X )
3 neg1cn 10651 . . . . . . 7  |-  -u 1  e.  CC
4 nvmdi.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 nvmdi.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
64, 5nvscl 25344 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
73, 6mp3an2 1312 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
873ad2antr3 1163 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( -u 1 S C )  e.  X )
91, 2, 83jca 1176 . . . 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 2467 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
114, 10, 5nvdi 25348 . . . 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 25347 . . . . . 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 1322 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
15143adantr2 1156 . . . 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 6311 . . 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 2508 . 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 25360 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  =  ( B ( +v
`  U ) (
-u 1 S C ) ) )
20193adant3r1 1205 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( B M C )  =  ( B ( +v `  U
) ( -u 1 S C ) ) )
2120oveq2d 6311 . 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 25344 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
24233adant3r3 1207 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S B )  e.  X )
254, 5nvscl 25344 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  C  e.  X )  ->  ( A S C )  e.  X )
26253adant3r2 1206 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S C )  e.  X )
274, 10, 5, 18nvmval 25360 . . 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 1228 . 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 2518 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 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505   -ucneg 9818   NrmCVeccnv 25300   +vcpv 25301   BaseSetcba 25302   .sOLDcns 25303   -vcnsb 25305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-sub 9819  df-neg 9820  df-grpo 25016  df-gid 25017  df-ginv 25018  df-gdiv 25019  df-ablo 25107  df-vc 25262  df-nv 25308  df-va 25311  df-ba 25312  df-sm 25313  df-0v 25314  df-vs 25315  df-nmcv 25316
This theorem is referenced by:  smcnlem  25430  minvecolem2  25614
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