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Theorem nvmdi 25972
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 1005 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  A  e.  CC )
2 simpr2 1006 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  B  e.  X )
3 neg1cn 10682 . . . . . . 7  |-  -u 1  e.  CC
4 nvmdi.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 nvmdi.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
64, 5nvscl 25948 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
73, 6mp3an2 1316 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
873ad2antr3 1166 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( -u 1 S C )  e.  X )
91, 2, 83jca 1179 . . . 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 2404 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
114, 10, 5nvdi 25952 . . . 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 25951 . . . . . 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 1326 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
15143adantr2 1159 . . . 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 6296 . . 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 2445 . 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 25964 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  =  ( B ( +v
`  U ) (
-u 1 S C ) ) )
20193adant3r1 1208 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( B M C )  =  ( B ( +v `  U
) ( -u 1 S C ) ) )
2120oveq2d 6296 . 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 25948 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
24233adant3r3 1210 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S B )  e.  X )
254, 5nvscl 25948 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  C  e.  X )  ->  ( A S C )  e.  X )
26253adant3r2 1209 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S C )  e.  X )
274, 10, 5, 18nvmval 25964 . . 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 1232 . 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 2455 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 976    = wceq 1407    e. wcel 1844   ` cfv 5571  (class class class)co 6280   CCcc 9522   1c1 9525   -ucneg 9844   NrmCVeccnv 25904   +vcpv 25905   BaseSetcba 25906   .sOLDcns 25907   -vcnsb 25909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-ltxr 9665  df-sub 9845  df-neg 9846  df-grpo 25620  df-gid 25621  df-ginv 25622  df-gdiv 25623  df-ablo 25711  df-vc 25866  df-nv 25912  df-va 25915  df-ba 25916  df-sm 25917  df-0v 25918  df-vs 25919  df-nmcv 25920
This theorem is referenced by:  smcnlem  26034  minvecolem2  26218
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