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Theorem clmmulg 18806
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 5988 . . . . 5  |-  ( x  =  0  ->  (
x  .xb  B )  =  ( 0  .xb  B ) )
2 oveq1 5988 . . . . 5  |-  ( x  =  0  ->  (
x  .x.  B )  =  ( 0  .x. 
B ) )
31, 2eqeq12d 2380 . . . 4  |-  ( x  =  0  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( 0 
.xb  B )  =  ( 0  .x.  B
) ) )
4 oveq1 5988 . . . . 5  |-  ( x  =  y  ->  (
x  .xb  B )  =  ( y  .xb  B ) )
5 oveq1 5988 . . . . 5  |-  ( x  =  y  ->  (
x  .x.  B )  =  ( y  .x.  B ) )
64, 5eqeq12d 2380 . . . 4  |-  ( x  =  y  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( y  .xb  B )  =  ( y  .x.  B ) ) )
7 oveq1 5988 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x  .xb  B )  =  ( ( y  +  1 )  .xb  B ) )
8 oveq1 5988 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x  .x.  B )  =  ( ( y  +  1 )  .x.  B ) )
97, 8eqeq12d 2380 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( (
y  +  1 ) 
.xb  B )  =  ( ( y  +  1 )  .x.  B
) ) )
10 oveq1 5988 . . . . 5  |-  ( x  =  -u y  ->  (
x  .xb  B )  =  ( -u y  .xb  B ) )
11 oveq1 5988 . . . . 5  |-  ( x  =  -u y  ->  (
x  .x.  B )  =  ( -u y  .x.  B ) )
1210, 11eqeq12d 2380 . . . 4  |-  ( x  =  -u y  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( -u y  .xb  B )  =  (
-u y  .x.  B
) ) )
13 oveq1 5988 . . . . 5  |-  ( x  =  A  ->  (
x  .xb  B )  =  ( A  .xb  B ) )
14 oveq1 5988 . . . . 5  |-  ( x  =  A  ->  (
x  .x.  B )  =  ( A  .x.  B ) )
1513, 14eqeq12d 2380 . . . 4  |-  ( x  =  A  ->  (
( x  .xb  B
)  =  ( x 
.x.  B )  <->  ( A  .xb 
B )  =  ( A  .x.  B ) ) )
16 clmmulg.1 . . . . . . 7  |-  V  =  ( Base `  W
)
17 eqid 2366 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
18 clmmulg.2 . . . . . . 7  |-  .xb  =  (.g
`  W )
1916, 17, 18mulg0 14782 . . . . . 6  |-  ( B  e.  V  ->  (
0  .xb  B )  =  ( 0g `  W ) )
2019adantl 452 . . . . 5  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .xb  B )  =  ( 0g `  W ) )
21 eqid 2366 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
22 clmmulg.3 . . . . . 6  |-  .x.  =  ( .s `  W )
2316, 21, 22, 17clm0vs 18803 . . . . 5  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .x.  B )  =  ( 0g `  W ) )
2420, 23eqtr4d 2401 . . . 4  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
0  .xb  B )  =  ( 0  .x. 
B ) )
25 oveq1 5988 . . . . . 6  |-  ( ( y  .xb  B )  =  ( y  .x.  B )  ->  (
( y  .xb  B
) ( +g  `  W
) B )  =  ( ( y  .x.  B ) ( +g  `  W ) B ) )
26 clmgrp 18781 . . . . . . . . . 10  |-  ( W  e. CMod  ->  W  e.  Grp )
27 grpmnd 14704 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
2826, 27syl 15 . . . . . . . . 9  |-  ( W  e. CMod  ->  W  e.  Mnd )
2928ad2antrr 706 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  W  e.  Mnd )
30 simpr 447 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  NN0 )
31 simplr 731 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  B  e.  V
)
32 eqid 2366 . . . . . . . . 9  |-  ( +g  `  W )  =  ( +g  `  W )
3316, 18, 32mulgnn0p1 14788 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  y  e.  NN0  /\  B  e.  V )  ->  (
( y  +  1 )  .xb  B )  =  ( ( y 
.xb  B ) ( +g  `  W ) B ) )
3429, 30, 31, 33syl3anc 1183 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .xb  B )  =  ( ( y  .xb  B
) ( +g  `  W
) B ) )
35 simpll 730 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  W  e. CMod )
36 eqid 2366 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3721, 36clmzss 18791 . . . . . . . . . . 11  |-  ( W  e. CMod  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
3837ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
39 nn0z 10197 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  y  e.  ZZ )
4039adantl 452 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  ZZ )
4138, 40sseldd 3267 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  y  e.  (
Base `  (Scalar `  W
) ) )
42 1z 10204 . . . . . . . . . . 11  |-  1  e.  ZZ
4342a1i 10 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  1  e.  ZZ )
4438, 43sseldd 3267 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  1  e.  (
Base `  (Scalar `  W
) ) )
4516, 21, 22, 36, 32clmvsdir 18801 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .x.  B )  =  ( ( y  .x.  B
) ( +g  `  W
) ( 1  .x. 
B ) ) )
4716, 22clmvs1 18802 . . . . . . . . . 10  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  (
1  .x.  B )  =  B )
4847adantr 451 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( 1  .x. 
B )  =  B )
4948oveq2d 5997 . . . . . . . 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 2398 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN0 )  ->  ( ( y  +  1 )  .x.  B )  =  ( ( y  .x.  B
) ( +g  `  W
) B ) )
5134, 50eqeq12d 2380 . . . . . 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 212 . . . . 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 423 . . . 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 5632 . . . . . 6  |-  ( ( y  .xb  B )  =  ( y  .x.  B )  ->  (
( inv g `  W ) `  (
y  .xb  B )
)  =  ( ( inv g `  W
) `  ( y  .x.  B ) ) )
5526ad2antrr 706 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  W  e.  Grp )
56 nnz 10196 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  ZZ )
5756adantl 452 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  y  e.  ZZ )
58 simplr 731 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  B  e.  V
)
59 eqid 2366 . . . . . . . . 9  |-  ( inv g `  W )  =  ( inv g `  W )
6016, 18, 59mulgneg 14795 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  y  e.  ZZ  /\  B  e.  V )  ->  ( -u y  .xb  B )  =  ( ( inv g `  W ) `
 ( y  .xb  B ) ) )
6155, 57, 58, 60syl3anc 1183 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( -u y  .xb  B )  =  ( ( inv g `  W ) `  (
y  .xb  B )
) )
62 simpll 730 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  W  e. CMod )
6337ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ZZ  C_  ( Base `  (Scalar `  W
) ) )
6463, 57sseldd 3267 . . . . . . . . 9  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  y  e.  (
Base `  (Scalar `  W
) ) )
6516, 21, 22, 59, 36, 62, 58, 64clmvsneg 18805 . . . . . . . 8  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( ( inv g `  W ) `
 ( y  .x.  B ) )  =  ( -u y  .x.  B ) )
6665eqcomd 2371 . . . . . . 7  |-  ( ( ( W  e. CMod  /\  B  e.  V )  /\  y  e.  NN )  ->  ( -u y  .x.  B )  =  ( ( inv g `  W ) `  (
y  .x.  B )
) )
6761, 66eqeq12d 2380 . . . . . 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 212 . . . . 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 423 . . . 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 10264 . . 3  |-  ( ( W  e. CMod  /\  B  e.  V )  ->  ( A  e.  ZZ  ->  ( A  .xb  B )  =  ( A  .x.  B ) ) )
71703impia 1149 . 2  |-  ( ( W  e. CMod  /\  B  e.  V  /\  A  e.  ZZ )  ->  ( A  .xb  B )  =  ( A  .x.  B
) )
72713com23 1158 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715    C_ wss 3238   ` cfv 5358  (class class class)co 5981   0cc0 8884   1c1 8885    + caddc 8887   -ucneg 9185   NNcn 9893   NN0cn0 10114   ZZcz 10175   Basecbs 13356   +g cplusg 13416  Scalarcsca 13419   .scvsca 13420   0gc0g 13610   Mndcmnd 14571   Grpcgrp 14572   inv gcminusg 14573  .gcmg 14576  CModcclm 18775
This theorem is referenced by:  clmzlmvsca  18809  minveclem2  19005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-seq 11211  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-mulg 14702  df-subg 14828  df-cmn 15301  df-mgp 15536  df-rng 15550  df-cring 15551  df-ur 15552  df-subrg 15753  df-lmod 15839  df-cnfld 16594  df-clm 18776
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