MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgnn0ass Unicode version

Theorem mulgnn0ass 14874
Description: Product of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgass.b  |-  B  =  ( Base `  G
)
mulgass.t  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
mulgnn0ass  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N 
.x.  X ) ) )

Proof of Theorem mulgnn0ass
StepHypRef Expression
1 simpll 731 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  G  e.  Mnd )
2 simprl 733 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  M  e.  NN )
3 simprr 734 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  N  e.  NN )
4 simpr3 965 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54adantr 452 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  X  e.  B )
6 mulgass.b . . . . . . 7  |-  B  =  ( Base `  G
)
7 mulgass.t . . . . . . 7  |-  .x.  =  (.g
`  G )
86, 7mulgnnass 14873 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B )
)  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N 
.x.  X ) ) )
91, 2, 3, 5, 8syl13anc 1186 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  -> 
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) ) )
109expr 599 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  ->  (
( M  x.  N
)  .x.  X )  =  ( M  .x.  ( N  .x.  X ) ) ) )
11 eqid 2404 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
126, 11, 7mulg0 14850 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
134, 12syl 16 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0 
.x.  X )  =  ( 0g `  G
) )
14 simpr1 963 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1514nn0cnd 10232 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  CC )
1615mul01d 9221 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  x.  0 )  =  0 )
1716oveq1d 6055 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( 0  .x.  X
) )
1813oveq2d 6056 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( M  .x.  ( 0g
`  G ) ) )
196, 7, 11mulgnn0z 14865 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  M  e.  NN0 )  -> 
( M  .x.  ( 0g `  G ) )  =  ( 0g `  G ) )
20193ad2antr1 1122 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0g `  G
) )  =  ( 0g `  G ) )
2118, 20eqtrd 2436 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( 0g `  G ) )
2213, 17, 213eqtr4d 2446 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) )
2322adantr 452 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) )
24 oveq2 6048 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
2524oveq1d 6055 . . . . . 6  |-  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( M  x.  0 )  .x.  X ) )
26 oveq1 6047 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
2726oveq2d 6056 . . . . . 6  |-  ( N  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( M  .x.  (
0  .x.  X )
) )
2825, 27eqeq12d 2418 . . . . 5  |-  ( N  =  0  ->  (
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) )  <->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) ) )
2923, 28syl5ibrcom 214 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( M  .x.  ( N  .x.  X ) ) ) )
30 simpr2 964 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
31 elnn0 10179 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3230, 31sylib 189 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3332adantr 452 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3410, 29, 33mpjaod 371 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( ( M  x.  N ) 
.x.  X )  =  ( M  .x.  ( N  .x.  X ) ) )
3534ex 424 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N 
.x.  X ) ) ) )
3630nn0cnd 10232 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  CC )
3736mul02d 9220 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0  x.  N )  =  0 )
3837oveq1d 6055 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  X
) )
396, 7mulgnn0cl 14861 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
40393adant3r1 1162 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
416, 11, 7mulg0 14850 . . . . 5  |-  ( ( N  .x.  X )  e.  B  ->  (
0  .x.  ( N  .x.  X ) )  =  ( 0g `  G
) )
4240, 41syl 16 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0 
.x.  ( N  .x.  X ) )  =  ( 0g `  G
) )
4313, 38, 423eqtr4d 2446 . . 3  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  ( N  .x.  X ) ) )
44 oveq1 6047 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
4544oveq1d 6055 . . . 4  |-  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( 0  x.  N )  .x.  X ) )
46 oveq1 6047 . . . 4  |-  ( M  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( 0  .x.  ( N  .x.  X ) ) )
4745, 46eqeq12d 2418 . . 3  |-  ( M  =  0  ->  (
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) )  <->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  ( N  .x.  X ) ) ) )
4843, 47syl5ibrcom 214 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( M  .x.  ( N  .x.  X ) ) ) )
49 elnn0 10179 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5014, 49sylib 189 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5135, 48, 50mpjaod 371 1  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N 
.x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   0cc0 8946    x. cmul 8951   NNcn 9956   NN0cn0 10177   Basecbs 13424   0gc0g 13678   Mndcmnd 14639  .gcmg 14644
This theorem is referenced by:  mulgass  14875  odmodnn0  15133
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-0g 13682  df-mnd 14645  df-mulg 14770
  Copyright terms: Public domain W3C validator