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Theorem mvecrtol2 25580
Description: Moving a vector from the right member of an equation into the left member. (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
mvecrtol2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1  =  ( V 2 + w V 3
)  <->  ( V1 - w V 2 )  =  V 3 ) )

Proof of Theorem mvecrtol2
StepHypRef Expression
1 vwit.2 . . . . . 6  |-  + w  =  ( 1st `  ( 2nd `  R ) )
21vecax1 25556 . . . . 5  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
3 ablogrpo 20967 . . . . 5  |-  ( + w  e.  AbelOp  ->  + w  e.  GrpOp )
42, 3syl 15 . . . 4  |-  ( R  e.  Vec  ->  + w  e.  GrpOp )
54adantr 451 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  + w  e.  GrpOp )
6 simpr1 961 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  V1  e.  W )
7 vwit.4 . . . . 5  |-  W  =  ran  + w
81, 7sum2vv 25565 . . . 4  |-  ( ( R  e.  Vec  /\  V 2  e.  W  /\  V 3  e.  W
)  ->  ( V 2 + w V 3
)  e.  W )
983adant3r1 1160 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 + w V 3 )  e.  W
)
10 simpr2 962 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  V 2  e.  W )
11 vwit.3 . . . 4  |-  - w  =  (  /g  `  + w )
127, 11grpodrcan 25478 . . 3  |-  ( ( + w  e.  GrpOp  /\  ( V1  e.  W  /\  ( V 2 + w V 3 )  e.  W  /\  V 2  e.  W ) )  -> 
( ( V1 - w V 2 )  =  ( ( V 2 + w V 3 ) - w V 2 )  <->  V1  =  ( V 2 + w V 3 ) ) )
135, 6, 9, 10, 12syl13anc 1184 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 - w V 2 )  =  ( ( V 2 + w V 3 ) - w V 2 )  <->  V1  =  ( V 2 + w V 3 ) ) )
141, 7addvecom 25569 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 + w V 3 )  =  ( V 3 + w V 2 ) )
15143adantr1 1114 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 + w V 3 )  =  ( V 3 + w V 2 ) )
1615oveq1d 5889 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V 2 + w V 3 ) - w V 2 )  =  ( ( V 3 + w V 2 ) - w V 2 )
)
17 simpl 443 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  R  e.  Vec  )
181, 7sum2vv 25565 . . . . . . . . . . . 12  |-  ( ( R  e.  Vec  /\  V 3  e.  W  /\  V 2  e.  W
)  ->  ( V 3 + w V 2
)  e.  W )
19183exp 1150 . . . . . . . . . . 11  |-  ( R  e.  Vec  ->  ( V 3  e.  W  ->  ( V 2  e.  W  ->  ( V 3 + w V 2 )  e.  W ) ) )
2019com3l 75 . . . . . . . . . 10  |-  ( V 3  e.  W  -> 
( V 2  e.  W  ->  ( R  e. 
Vec  ->  ( V 3 + w V 2 )  e.  W ) ) )
2120impcom 419 . . . . . . . . 9  |-  ( ( V 2  e.  W  /\  V 3  e.  W
)  ->  ( R  e.  Vec  ->  ( V 3 + w V 2
)  e.  W ) )
22213adant1 973 . . . . . . . 8  |-  ( (
V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
)  ->  ( R  e.  Vec  ->  ( V 3 + w V 2
)  e.  W ) )
2322impcom 419 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w V 2 )  e.  W
)
24 vwit.1 . . . . . . . 8  |-  0 w  =  (GId `  + w )
25 eqid 2296 . . . . . . . 8  |-  ( inv `  + w )  =  ( inv `  + w )
2624, 1, 11, 7, 25sub2vec 25575 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( ( V 3 + w V 2 )  e.  W  /\  V 2  e.  W ) )  -> 
( ( V 3 + w V 2 ) - w V 2 )  =  ( ( V 3 + w V 2 ) + w
( ( inv `  + w ) `  V 2 ) ) )
2717, 23, 10, 26syl12anc 1180 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V 3 + w V 2 ) - w V 2 )  =  ( ( V 3 + w V 2 ) + w ( ( inv `  + w ) `  V 2 ) ) )
28 simpr3 963 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  V 3  e.  W )
291rneqi 4921 . . . . . . . . . . 11  |-  ran  + w  =  ran  ( 1st `  ( 2nd `  R
) )
307, 29eqtri 2316 . . . . . . . . . 10  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
311fveq2i 5544 . . . . . . . . . 10  |-  ( inv `  + w )  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
3230, 31claddinvvec 25563 . . . . . . . . 9  |-  ( ( R  e.  Vec  /\  V 2  e.  W
)  ->  ( ( inv `  + w ) `  V 2 )  e.  W )
33323ad2antr2 1121 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( inv `  + w ) `  V 2 )  e.  W
)
3428, 10, 333jca 1132 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3  e.  W  /\  V 2  e.  W  /\  ( ( inv `  + w ) `  V 2 )  e.  W
) )
351, 7addvecass 25568 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V 3  e.  W  /\  V 2  e.  W  /\  ( ( inv `  + w ) `  V 2 )  e.  W
) )  ->  ( V 3 + w ( V 2 + w (
( inv `  + w ) `  V 2 ) ) )  =  ( ( V 3 + w V 2 ) + w
( ( inv `  + w ) `  V 2 ) ) )
3634, 35syldan 456 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w ( V 2 + w (
( inv `  + w ) `  V 2 ) ) )  =  ( ( V 3 + w V 2 ) + w
( ( inv `  + w ) `  V 2 ) ) )
37 simp2 956 . . . . . . . . . 10  |-  ( (
V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
)  ->  V 2  e.  W )
3837, 37jca 518 . . . . . . . . 9  |-  ( (
V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
)  ->  ( V 2  e.  W  /\  V 2  e.  W
) )
3924, 1, 11, 7, 25sub2vec 25575 . . . . . . . . 9  |-  ( ( R  e.  Vec  /\  ( V 2  e.  W  /\  V 2  e.  W
) )  ->  ( V 2 - w V 2 )  =  ( V 2 + w
( ( inv `  + w ) `  V 2 ) ) )
4038, 39sylan2 460 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 - w V 2 )  =  ( V 2 + w
( ( inv `  + w ) `  V 2 ) ) )
4140eqcomd 2301 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 + w (
( inv `  + w ) `  V 2 ) )  =  ( V 2 - w V 2 ) )
4241oveq2d 5890 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w ( V 2 + w (
( inv `  + w ) `  V 2 ) ) )  =  ( V 3 + w ( V 2 - w V 2 )
) )
4327, 36, 423eqtr2d 2334 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V 3 + w V 2 ) - w V 2 )  =  ( V 3 + w ( V 2 - w V 2 )
) )
4424, 1, 11, 7vwit 25574 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  V 2  e.  W
)  ->  ( V 2 - w V 2
)  =  0 w
)
45443ad2antr2 1121 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 2 - w V 2 )  =  0 w )
4645oveq2d 5890 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w ( V 2 - w V 2 ) )  =  ( V 3 + w 0 w )
)
471, 7, 24addnull1 25566 . . . . . . 7  |-  ( ( R  e.  Vec  /\  V 3  e.  W
)  ->  ( V 3 + w 0 w
)  =  V 3
)
48473ad2antr3 1122 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w 0 w )  =  V 3 )
4946, 48eqtrd 2328 . . . . 5  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V 3 + w ( V 2 - w V 2 ) )  =  V 3 )
5043, 49eqtrd 2328 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V 3 + w V 2 ) - w V 2 )  =  V 3 )
5116, 50eqtrd 2328 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V 2 + w V 3 ) - w V 2 )  =  V 3 )
5251eqeq2d 2307 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  (
( V1 - w V 2 )  =  ( ( V 2 + w V 3 ) - w V 2 )  <->  ( V1 - w V 2 )  =  V 3 ) )
5313, 52bitr3d 246 1  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W
) )  ->  ( V1  =  ( V 2 + w V 3
)  <->  ( V1 - w V 2 )  =  V 3 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871    /g cgs 20872   AbelOpcablo 20964    Vec cvec 25552
This theorem is referenced by:  mulveczer  25582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vec 25553
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