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Theorem clmmulg 22901
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 6657 . . . . 5 |- ( x = 0 -> ( x .xb B ) = ( 0 .xb B ) )
2 oveq1 6657 . . . . 5 |- ( x = 0 -> ( x .x. B ) = ( 0 .x. B ) )
31, 2eqeq12d 2637 . . . 4 |- ( x = 0 -> ( ( x .xb B ) = ( x .x. B ) <-> ( 0 .xb B ) = ( 0 .x. B ) ) )
4 oveq1 6657 . . . . 5 |- ( x = y -> ( x .xb B ) = ( y .xb B ) )
5 oveq1 6657 . . . . 5 |- ( x = y -> ( x .x. B ) = ( y .x. B ) )
64, 5eqeq12d 2637 . . . 4 |- ( x = y -> ( ( x .xb B ) = ( x .x. B ) <-> ( y .xb B ) = ( y .x. B ) ) )
7 oveq1 6657 . . . . 5 |- ( x = ( y + 1 ) -> ( x .xb B ) = ( ( y + 1 ) .xb B ) )
8 oveq1 6657 . . . . 5 |- ( x = ( y + 1 ) -> ( x .x. B ) = ( ( y + 1 ) .x. B ) )
97, 8eqeq12d 2637 . . . 4 |- ( x = ( y + 1 ) -> ( ( x .xb B ) = ( x .x. B ) <-> ( ( y + 1 ) .xb B ) = ( ( y + 1 ) .x. B ) ) )
10 oveq1 6657 . . . . 5 |- ( x = -u y -> ( x .xb B ) = ( -u y .xb B ) )
11 oveq1 6657 . . . . 5 |- ( x = -u y -> ( x .x. B ) = ( -u y .x. B ) )
1210, 11eqeq12d 2637 . . . 4 |- ( x = -u y -> ( ( x .xb B ) = ( x .x. B ) <-> ( -u y .xb B ) = ( -u y .x. B ) ) )
13 oveq1 6657 . . . . 5 |- ( x = A -> ( x .xb B ) = ( A .xb B ) )
14 oveq1 6657 . . . . 5 |- ( x = A -> ( x .x. B ) = ( A .x. B ) )
1513, 14eqeq12d 2637 . . . 4 |- ( x = A -> ( ( x .xb B ) = ( x .x. B ) <-> ( A .xb B ) = ( A .x. B ) ) )
16 clmmulg.1 . . . . . . 7 |- V = ( Base ` W )
17 eqid 2622 . . . . . . 7 |- ( 0g ` W ) = ( 0g ` W )
18 clmmulg.2 . . . . . . 7 |- .xb = (.g ` W )
1916, 17, 18mulg0 17546 . . . . . 6 |- ( B e. V -> ( 0 .xb B ) = ( 0g ` W ) )
2019adantl 482 . . . . 5 |- ( ( W e. CMod /\ B e. V ) -> ( 0 .xb B ) = ( 0g ` W ) )
21 eqid 2622 . . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
22 clmmulg.3 . . . . . 6 |- .x. = ( .s ` W )
2316, 21, 22, 17clm0vs 22895 . . . . 5 |- ( ( W e. CMod /\ B e. V ) -> ( 0 .x. B ) = ( 0g ` W ) )
2420, 23eqtr4d 2659 . . . 4 |- ( ( W e. CMod /\ B e. V ) -> ( 0 .xb B ) = ( 0 .x. B ) )
25 oveq1 6657 . . . . . 6 |- ( ( y .xb B ) = ( y .x. B ) -> ( ( y .xb B ) ( +g ` W ) B ) = ( ( y .x. B ) ( +g ` W ) B ) )
26 clmgrp 22868 . . . . . . . . . 10 |- ( W e. CMod -> W e. Grp )
27 grpmnd 17429 . . . . . . . . . 10 |- ( W e. Grp -> W e. Mnd )
2826, 27syl 17 . . . . . . . . 9 |- ( W e. CMod -> W e. Mnd )
2928ad2antrr 762 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> W e. Mnd )
30 simpr 477 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> y e. NN0 )
31 simplr 792 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> B e. V )
32 eqid 2622 . . . . . . . . 9 |- ( +g ` W ) = ( +g ` W )
3316, 18, 32mulgnn0p1 17552 . . . . . . . 8 |- ( ( W e. Mnd /\ y e. NN0 /\ B e. V ) -> ( ( y + 1 ) .xb B ) = ( ( y .xb B ) ( +g ` W ) B ) )
3429, 30, 31, 33syl3anc 1326 . . . . . . 7 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( ( y + 1 ) .xb B ) = ( ( y .xb B ) ( +g ` W ) B ) )
35 simpll 790 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> W e. CMod )
36 eqid 2622 . . . . . . . . . . . 12 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
3721, 36clmzss 22878 . . . . . . . . . . 11 |- ( W e. CMod -> ZZ C_ ( Base ` (Scalar ` W ) ) )
3837ad2antrr 762 . . . . . . . . . 10 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ZZ C_ ( Base ` (Scalar ` W ) ) )
39 nn0z 11400 . . . . . . . . . . 11 |- ( y e. NN0 -> y e. ZZ )
4039adantl 482 . . . . . . . . . 10 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> y e. ZZ )
4138, 40sseldd 3604 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> y e. ( Base ` (Scalar ` W ) ) )
42 1zzd 11408 . . . . . . . . . 10 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> 1 e. ZZ )
4338, 42sseldd 3604 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> 1 e. ( Base ` (Scalar ` W ) ) )
4416, 21, 22, 36, 32clmvsdir 22891 . . . . . . . . 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 ) ) )
4535, 41, 43, 31, 44syl13anc 1328 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( ( y + 1 ) .x. B ) = ( ( y .x. B ) ( +g ` W ) ( 1 .x. B ) ) )
4616, 22clmvs1 22893 . . . . . . . . . 10 |- ( ( W e. CMod /\ B e. V ) -> ( 1 .x. B ) = B )
4746adantr 481 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( 1 .x. B ) = B )
4847oveq2d 6666 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( ( y .x. B ) ( +g ` W ) ( 1 .x. B ) ) = ( ( y .x. B ) ( +g ` W ) B ) )
4945, 48eqtrd 2656 . . . . . . 7 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( ( y + 1 ) .x. B ) = ( ( y .x. B ) ( +g ` W ) B ) )
5034, 49eqeq12d 2637 . . . . . 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 ) ) )
5125, 50syl5ibr 236 . . . . 5 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN0 ) -> ( ( y .xb B ) = ( y .x. B ) -> ( ( y + 1 ) .xb B ) = ( ( y + 1 ) .x. B ) ) )
5251ex 450 . . . 4 |- ( ( W e. CMod /\ B e. V ) -> ( y e. NN0 -> ( ( y .xb B ) = ( y .x. B ) -> ( ( y + 1 ) .xb B ) = ( ( y + 1 ) .x. B ) ) ) )
53 fveq2 6191 . . . . . 6 |- ( ( y .xb B ) = ( y .x. B ) -> ( ( invg ` W ) ` ( y .xb B ) ) = ( ( invg ` W ) ` ( y .x. B ) ) )
5426ad2antrr 762 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> W e. Grp )
55 nnz 11399 . . . . . . . . 9 |- ( y e. NN -> y e. ZZ )
5655adantl 482 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> y e. ZZ )
57 simplr 792 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> B e. V )
58 eqid 2622 . . . . . . . . 9 |- ( invg ` W ) = ( invg ` W )
5916, 18, 58mulgneg 17560 . . . . . . . 8 |- ( ( W e. Grp /\ y e. ZZ /\ B e. V ) -> ( -u y .xb B ) = ( ( invg ` W ) ` ( y .xb B ) ) )
6054, 56, 57, 59syl3anc 1326 . . . . . . 7 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ( -u y .xb B ) = ( ( invg ` W ) ` ( y .xb B ) ) )
61 simpll 790 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> W e. CMod )
6237ad2antrr 762 . . . . . . . . . 10 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ZZ C_ ( Base ` (Scalar ` W ) ) )
6362, 56sseldd 3604 . . . . . . . . 9 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> y e. ( Base ` (Scalar ` W ) ) )
6416, 21, 22, 58, 36, 61, 57, 63clmvsneg 22900 . . . . . . . 8 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ( ( invg ` W ) ` ( y .x. B ) ) = ( -u y .x. B ) )
6564eqcomd 2628 . . . . . . 7 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ( -u y .x. B ) = ( ( invg ` W ) ` ( y .x. B ) ) )
6660, 65eqeq12d 2637 . . . . . 6 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ( ( -u y .xb B ) = ( -u y .x. B ) <-> ( ( invg ` W ) ` ( y .xb B ) ) = ( ( invg ` W ) ` ( y .x. B ) ) ) )
6753, 66syl5ibr 236 . . . . 5 |- ( ( ( W e. CMod /\ B e. V ) /\ y e. NN ) -> ( ( y .xb B ) = ( y .x. B ) -> ( -u y .xb B ) = ( -u y .x. B ) ) )
6867ex 450 . . . 4 |- ( ( W e. CMod /\ B e. V ) -> ( y e. NN -> ( ( y .xb B ) = ( y .x. B ) -> ( -u y .xb B ) = ( -u y .x. B ) ) ) )
693, 6, 9, 12, 15, 24, 52, 68zindd 11478 . . 3 |- ( ( W e. CMod /\ B e. V ) -> ( A e. ZZ -> ( A .xb B ) = ( A .x. B ) ) )
70693impia 1261 . 2 |- ( ( W e. CMod /\ B e. V /\ A e. ZZ ) -> ( A .xb B ) = ( A .x. B ) )
71703com23 1271 1 |- ( ( W e. CMod /\ A e. ZZ /\ B e. V ) -> ( A .xb B ) = ( A .x. B ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Mndcmnd 17294   Grpcgrp 17422   invgcminusg 17423  .gcmg 17540  CModcclm 22862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-cnfld 19747  df-clm 22863
This theorem is referenced by:  clmzlmvsca  22913  minveclem2  23197
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