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Theorem clmmulg 19075
Description: The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
clmmulg.1  |-  V  =  ( Base `  W
)
clmmulg.2  |-  .xb  =  (.g
`  W )
clmmulg.3  |-  .x.  =  ( .s `  W )
Assertion
Ref Expression
clmmulg  |-  ( ( W  e. CMod  /\  A  e.  ZZ  /\  B  e.  V )  ->  ( A  .xb  B )  =  ( A  .x.  B
) )

Proof of Theorem clmmulg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6051 . . . . 5  |-  ( x  =  0  ->  (
x  .xb  B )  =  ( 0  .xb  B ) )
2 oveq1 6051 . . . . 5  |-  ( x  =  0  ->  (
x  .x.  B )  =  ( 0  .x. 
B ) )
31, 2eqeq12d 2422 . . . 4  |-  ( x  =  0  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( 0 
.xb  B )  =  ( 0  .x.  B
) ) )
4 oveq1 6051 . . . . 5  |-  ( x  =  y  ->  (
x  .xb  B )  =  ( y  .xb  B ) )
5 oveq1 6051 . . . . 5  |-  ( x  =  y  ->  (
x  .x.  B )  =  ( y  .x.  B ) )
64, 5eqeq12d 2422 . . . 4  |-  ( x  =  y  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( y  .xb  B )  =  ( y  .x.  B ) ) )
7 oveq1 6051 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x  .xb  B )  =  ( ( y  +  1 )  .xb  B ) )
8 oveq1 6051 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x  .x.  B )  =  ( ( y  +  1 )  .x.  B ) )
97, 8eqeq12d 2422 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( (
y  +  1 ) 
.xb  B )  =  ( ( y  +  1 )  .x.  B
) ) )
10 oveq1 6051 . . . . 5  |-  ( x  =  -u y  ->  (
x  .xb  B )  =  ( -u y  .xb  B ) )
11 oveq1 6051 . . . . 5  |-  ( x  =  -u y  ->  (
x  .x.  B )  =  ( -u y  .x.  B ) )
1210, 11eqeq12d 2422 . . . 4  |-  ( x  =  -u y  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( -u y  .xb  B )  =  (
-u y  .x.  B
) ) )
13 oveq1 6051 . . . . 5  |-  ( x  =  A  ->  (
x  .xb  B )  =  ( A  .xb  B ) )
14 oveq1 6051 . . . . 5  |-  ( x  =  A  ->  (
x  .x.  B )  =  ( A  .x.  B ) )
1513, 14eqeq12d 2422 . . . 4  |-  ( x  =  A  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( A  .xb 
B )  =  ( A  .x.  B ) ) )
16 clmmulg.1 . . . . . . 7  |-  V  =  ( Base `  W
)
17 eqid 2408 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
18 clmmulg.2 . . . . . . 7  |-  .xb  =  (.g
`  W )
1916, 17, 18mulg0 14854 . . . . . 6  |-  ( B  e.  V  ->  (
0  .xb  B )  =  ( 0g `  W ) )
2019adantl 453 . . . . 5  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .xb  B )  =  ( 0g `  W ) )
21 eqid 2408 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
22 clmmulg.3 . . . . . 6  |-  .x.  =  ( .s `  W )
2316, 21, 22, 17clm0vs 19072 . . . . 5  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .x.  B )  =  ( 0g `  W ) )
2420, 23eqtr4d 2443 . . . 4  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .xb  B )  =  ( 0  .x. 
B ) )
25 oveq1 6051 . . . . . 6  |-  ( ( y  .xb  B )  =  ( y  .x.  B )  ->  (
( y  .xb  B
) ( +g  `  W
) B )  =  ( ( y  .x.  B ) ( +g  `  W ) B ) )
26 clmgrp 19050 . . . . . . . . . 10  |-  ( W  e. CMod  ->  W  e.  Grp )
27 grpmnd 14776 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
2826, 27syl 16 . . . . . . . . 9  |-  ( W  e. CMod  ->  W  e.  Mnd )
2928ad2antrr 707 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  W  e.  Mnd )
30 simpr 448 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  NN0 )
31 simplr 732 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  B  e.  V
)
32 eqid 2408 . . . . . . . . 9  |-  ( +g  `  W )  =  ( +g  `  W )
3316, 18, 32mulgnn0p1 14860 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  y  e.  NN0  /\  B  e.  V )  ->  (
( y  +  1 )  .xb  B )  =  ( ( y 
.xb  B ) ( +g  `  W ) B ) )
3429, 30, 31, 33syl3anc 1184 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .xb  B )  =  ( ( y  .xb  B
) ( +g  `  W
) B ) )
35 simpll 731 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  W  e. CMod )
36 eqid 2408 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3721, 36clmzss 19060 . . . . . . . . . . 11  |-  ( W  e. CMod  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
3837ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
39 nn0z 10264 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  y  e.  ZZ )
4039adantl 453 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  ZZ )
4138, 40sseldd 3313 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  (
Base `  (Scalar `  W
) ) )
42 1z 10271 . . . . . . . . . . 11  |-  1  e.  ZZ
4342a1i 11 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  1  e.  ZZ )
4438, 43sseldd 3313 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  1  e.  (
Base `  (Scalar `  W
) ) )
4516, 21, 22, 36, 32clmvsdir 19070 . . . . . . . . 9  |-  ( ( W  e. CMod  /\  (
y  e.  ( Base `  (Scalar `  W )
)  /\  1  e.  ( Base `  (Scalar `  W
) )  /\  B  e.  V ) )  -> 
( ( y  +  1 )  .x.  B
)  =  ( ( y  .x.  B ) ( +g  `  W
) ( 1  .x. 
B ) ) )
4635, 41, 44, 31, 45syl13anc 1186 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .x.  B )  =  ( ( y  .x.  B
) ( +g  `  W
) ( 1  .x. 
B ) ) )
4716, 22clmvs1 19071 . . . . . . . . . 10  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
1  .x.  B )  =  B )
4847adantr 452 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( 1  .x. 
B )  =  B )
4948oveq2d 6060 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y 
.x.  B ) ( +g  `  W ) ( 1  .x.  B
) )  =  ( ( y  .x.  B
) ( +g  `  W
) B ) )
5046, 49eqtrd 2440 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .x.  B )  =  ( ( y  .x.  B
) ( +g  `  W
) B ) )
5134, 50eqeq12d 2422 . . . . . 6  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( ( y  +  1 ) 
.xb  B )  =  ( ( y  +  1 )  .x.  B
)  <->  ( ( y 
.xb  B ) ( +g  `  W ) B )  =  ( ( y  .x.  B
) ( +g  `  W
) B ) ) )
5225, 51syl5ibr 213 . . . . 5  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y 
.xb  B )  =  ( y  .x.  B
)  ->  ( (
y  +  1 ) 
.xb  B )  =  ( ( y  +  1 )  .x.  B
) ) )
5352ex 424 . . . 4  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
y  e.  NN0  ->  ( ( y  .xb  B
)  =  ( y 
.x.  B )  -> 
( ( y  +  1 )  .xb  B
)  =  ( ( y  +  1 ) 
.x.  B ) ) ) )
54 fveq2 5691 . . . . . 6  |-  ( ( y  .xb  B )  =  ( y  .x.  B )  ->  (
( inv g `  W ) `  (
y  .xb  B )
)  =  ( ( inv g `  W
) `  ( y  .x.  B ) ) )
5526ad2antrr 707 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  W  e.  Grp )
56 nnz 10263 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  ZZ )
5756adantl 453 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  y  e.  ZZ )
58 simplr 732 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  B  e.  V
)
59 eqid 2408 . . . . . . . . 9  |-  ( inv g `  W )  =  ( inv g `  W )
6016, 18, 59mulgneg 14867 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  y  e.  ZZ  /\  B  e.  V )  ->  ( -u y  .xb  B )  =  ( ( inv g `  W ) `
 ( y  .xb  B ) ) )
6155, 57, 58, 60syl3anc 1184 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( -u y  .xb  B )  =  ( ( inv g `  W ) `  (
y  .xb  B )
) )
62 simpll 731 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  W  e. CMod )
6337ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
6463, 57sseldd 3313 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  y  e.  (
Base `  (Scalar `  W
) ) )
6516, 21, 22, 59, 36, 62, 58, 64clmvsneg 19074 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( ( inv g `  W ) `
 ( y  .x.  B ) )  =  ( -u y  .x.  B ) )
6665eqcomd 2413 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( -u y  .x.  B )  =  ( ( inv g `  W ) `  (
y  .x.  B )
) )
6761, 66eqeq12d 2422 . . . . . 6  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( ( -u y  .xb  B )  =  ( -u y  .x.  B )  <->  ( ( inv g `  W ) `
 ( y  .xb  B ) )  =  ( ( inv g `  W ) `  (
y  .x.  B )
) ) )
6854, 67syl5ibr 213 . . . . 5  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( ( y 
.xb  B )  =  ( y  .x.  B
)  ->  ( -u y  .xb  B )  =  (
-u y  .x.  B
) ) )
6968ex 424 . . . 4  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
y  e.  NN  ->  ( ( y  .xb  B
)  =  ( y 
.x.  B )  -> 
( -u y  .xb  B
)  =  ( -u y  .x.  B ) ) ) )
703, 6, 9, 12, 15, 24, 53, 69zindd 10331 . . 3  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  ( A  e.  ZZ  ->  ( A  .xb  B )  =  ( A  .x.  B ) ) )
71703impia 1150 . 2  |-  ( ( W  e. CMod  /\  B  e.  V  /\  A  e.  ZZ )  ->  ( A  .xb  B )  =  ( A  .x.  B
) )
72713com23 1159 1  |-  ( ( W  e. CMod  /\  A  e.  ZZ  /\  B  e.  V )  ->  ( A  .xb  B )  =  ( A  .x.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3284   ` cfv 5417  (class class class)co 6044   0cc0 8950   1c1 8951    + caddc 8953   -ucneg 9252   NNcn 9960   NN0cn0 10181   ZZcz 10242   Basecbs 13428   +g cplusg 13488  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   Mndcmnd 14643   Grpcgrp 14644   inv gcminusg 14645  .gcmg 14648  CModcclm 19044
This theorem is referenced by:  clmzlmvsca  19078  minveclem2  19284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-seq 11283  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-starv 13503  df-tset 13507  df-ple 13508  df-ds 13510  df-unif 13511  df-0g 13686  df-mnd 14649  df-grp 14771  df-minusg 14772  df-mulg 14774  df-subg 14900  df-cmn 15373  df-mgp 15608  df-rng 15622  df-cring 15623  df-ur 15624  df-subrg 15825  df-lmod 15911  df-cnfld 16663  df-clm 19045
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