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Theorem dblsubvec 25474
Description: Double subtraction of vectors. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
vwit.1  |-  0 w  =  (GId `  + w )
vwit.2  |-  + w  =  ( 1st `  ( 2nd `  R ) )
vwit.3  |-  - w  =  (  /g  `  + w )
vwit.4  |-  W  =  ran  + w
Assertion
Ref Expression
dblsubvec  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 - w V 2 ) - w V 3 )  =  (
V1 - w ( V 2 + w V 3 ) ) )

Proof of Theorem dblsubvec
StepHypRef Expression
1 vwit.1 . . . . 5  |-  0 w  =  (GId `  + w )
2 vwit.2 . . . . 5  |-  + w  =  ( 1st `  ( 2nd `  R ) )
3 vwit.3 . . . . 5  |-  - w  =  (  /g  `  + w )
4 vwit.4 . . . . 5  |-  W  =  ran  + w
5 eqid 2283 . . . . 5  |-  ( inv `  + w )  =  ( inv `  + w )
61, 2, 3, 4, 5sub2vec 25472 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( V1 - w V 2
)  =  ( V1 + w ( ( inv `  + w ) `  V 2 ) ) )
763adantr3 1116 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1 - w V 2
)  =  ( V1 + w ( ( inv `  + w ) `  V 2 ) ) )
87oveq1d 5873 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 - w V 2 ) - w V 3 )  =  ( ( V1 + w
( ( inv `  + w ) `  V 2 ) ) - w V 3 ) )
9 simpl 443 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  R  e.  Vec  )
10 simpr1 961 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  V1  e.  W )
112rneqi 4905 . . . . . . 7  |-  ran  + w  =  ran  ( 1st `  ( 2nd `  R
) )
124, 11eqtri 2303 . . . . . 6  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
132fveq2i 5528 . . . . . 6  |-  ( inv `  + w )  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
1412, 13claddinvvec 25460 . . . . 5  |-  ( ( R  e.  Vec  /\  V 2  e.  W
)  ->  ( ( inv `  + w ) `  V 2 )  e.  W )
15143ad2antr2 1121 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( inv `  + w ) `  V 2 )  e.  W
)
162, 4sum2vv 25462 . . . 4  |-  ( ( R  e.  Vec  /\  V1  e.  W  /\  (
( inv `  + w ) `  V 2 )  e.  W
)  ->  ( V1 + w ( ( inv `  + w ) `  V 2 ) )  e.  W )
179, 10, 15, 16syl3anc 1182 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1 + w ( ( inv `  + w
) `  V 2 )
)  e.  W )
18 simpr3 963 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  V 3  e.  W )
191, 2, 3, 4, 5sub2vec 25472 . . 3  |-  ( ( R  e.  Vec  /\  ( ( V1 + w ( ( inv `  + w ) `  V 2 ) )  e.  W  /\  V 3  e.  W ) )  -> 
( ( V1 + w ( ( inv `  + w ) `  V 2 ) ) - w V 3 )  =  ( ( V1 + w ( ( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) ) )
209, 17, 18, 19syl12anc 1180 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 + w (
( inv `  + w ) `  V 2 ) ) - w V 3 )  =  ( ( V1 + w ( ( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) ) )
212, 4, 5invaddvec 25467 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( inv `  + w ) `  ( V 2 + w V 3 ) )  =  ( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) )
22213adantr1 1114 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( inv `  + w ) `  ( V 2 + w V 3 ) )  =  ( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) )
2322eqcomd 2288 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) )  =  ( ( inv `  + w ) `  ( V 2 + w V 3 ) ) )
2423oveq2d 5874 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1 + w ( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) )  =  ( V1 + w ( ( inv `  + w ) `  ( V 2 + w V 3 ) ) ) )
2512, 13claddinvvec 25460 . . . . . 6  |-  ( ( R  e.  Vec  /\  V 3  e.  W
)  ->  ( ( inv `  + w ) `  V 3 )  e.  W )
26253ad2antr3 1122 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( inv `  + w ) `  V 3 )  e.  W
)
2710, 15, 263jca 1132 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1  e.  W  /\  (
( inv `  + w ) `  V 2 )  e.  W  /\  ( ( inv `  + w ) `  V 3 )  e.  W
) )
282, 4addvecass 25465 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  ( ( inv `  + w ) `  V 2 )  e.  W  /\  ( ( inv `  + w ) `  V 3 )  e.  W
) )  ->  ( V1 + w ( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) )  =  ( ( V1 + w ( ( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) ) )
2928eqcomd 2288 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  ( ( inv `  + w ) `  V 2 )  e.  W  /\  ( ( inv `  + w ) `  V 3 )  e.  W
) )  ->  (
( V1 + w (
( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) )  =  ( V1 + w
( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) ) )
3027, 29syldan 456 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 + w (
( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) )  =  ( V1 + w
( ( ( inv `  + w ) `  V 2 ) + w
( ( inv `  + w ) `  V 3 ) ) ) )
312, 4sum2vv 25462 . . . . 5  |-  ( ( R  e.  Vec  /\  V 2  e.  W  /\  V 3  e.  W
)  ->  ( V 2 + w V 3
)  e.  W )
32313adant3r1 1160 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 + w V 3 )  e.  W
)
331, 2, 3, 4, 5sub2vec 25472 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  ( V 2 + w V 3 )  e.  W
) )  ->  ( V1 - w ( V 2 + w V 3 ) )  =  ( V1 + w
( ( inv `  + w ) `  ( V 2 + w V 3 ) ) ) )
349, 10, 32, 33syl12anc 1180 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1 - w ( V 2 + w V 3 ) )  =  ( V1 + w
( ( inv `  + w ) `  ( V 2 + w V 3 ) ) ) )
3524, 30, 343eqtr4d 2325 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 + w (
( inv `  + w ) `  V 2 ) ) + w ( ( inv `  + w ) `  V 3 ) )  =  ( V1 - w
( V 2 + w V 3 ) ) )
368, 20, 353eqtrd 2319 1  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 - w V 2 ) - w V 3 )  =  (
V1 - w ( V 2 + w V 3 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   invcgn 20855    /g cgs 20856    Vec cvec 25449
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-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-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vec 25450
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