MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvmdi Unicode version

Theorem nvmdi 21208
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 961 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  A  e.  CC )
2 simpr2 962 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  B  e.  X )
3 neg1cn 9813 . . . . . . 7  |-  -u 1  e.  CC
4 nvmdi.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 nvmdi.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
64, 5nvscl 21184 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
73, 6mp3an2 1265 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X )
873ad2antr3 1122 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( -u 1 S C )  e.  X )
91, 2, 83jca 1132 . . . 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 2283 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
114, 10, 5nvdi 21188 . . . 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 456 . . 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 21187 . . . . . 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 1275 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  C  e.  X ) )  -> 
( A S (
-u 1 S C ) )  =  (
-u 1 S ( A S C ) ) )
15143adantr2 1115 . . . 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 5874 . . 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 2315 . 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 21200 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  =  ( B ( +v
`  U ) (
-u 1 S C ) ) )
20193adant3r1 1160 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( B M C )  =  ( B ( +v `  U
) ( -u 1 S C ) ) )
2120oveq2d 5874 . 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 443 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  U  e.  NrmCVec )
234, 5nvscl 21184 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
24233adant3r3 1162 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S B )  e.  X )
254, 5nvscl 21184 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  C  e.  X )  ->  ( A S C )  e.  X )
26253adant3r2 1161 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S C )  e.  X )
274, 10, 5, 18nvmval 21200 . . 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 1182 . 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 2325 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   -vcnsb 21145
This theorem is referenced by:  smcnlem  21270  minvecolem2  21454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156
  Copyright terms: Public domain W3C validator