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

Theorem mulgnn0ass 15649
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 748 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  G  e.  Mnd )
2 simprl 750 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  M  e.  NN )
3 simprr 751 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  N  e.  NN )
4 simpr3 991 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54adantr 462 . . . . . 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 15648 . . . . . 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 1215 . . . . 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 612 . . . 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 2441 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
126, 11, 7mulg0 15625 . . . . . . . 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 989 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1514nn0cnd 10634 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  CC )
1615mul01d 9564 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  x.  0 )  =  0 )
1716oveq1d 6105 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( 0  .x.  X
) )
1813oveq2d 6106 . . . . . . . 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 15640 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  M  e.  NN0 )  -> 
( M  .x.  ( 0g `  G ) )  =  ( 0g `  G ) )
20193ad2antr1 1148 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0g `  G
) )  =  ( 0g `  G ) )
2118, 20eqtrd 2473 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( 0g `  G ) )
2213, 17, 213eqtr4d 2483 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) )
2322adantr 462 . . . . 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 6098 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
2524oveq1d 6105 . . . . . 6  |-  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( M  x.  0 )  .x.  X ) )
26 oveq1 6097 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
2726oveq2d 6106 . . . . . 6  |-  ( N  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( M  .x.  (
0  .x.  X )
) )
2825, 27eqeq12d 2455 . . . . 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 222 . . . 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 990 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
31 elnn0 10577 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3230, 31sylib 196 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3332adantr 462 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3410, 29, 33mpjaod 381 . . 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 434 . 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 10634 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  CC )
3736mul02d 9563 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0  x.  N )  =  0 )
3837oveq1d 6105 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  X
) )
396, 7mulgnn0cl 15636 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
40393adant3r1 1191 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
416, 11, 7mulg0 15625 . . . . 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 2483 . . 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 6097 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
4544oveq1d 6105 . . . 4  |-  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( 0  x.  N )  .x.  X ) )
46 oveq1 6097 . . . 4  |-  ( M  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( 0  .x.  ( N  .x.  X ) ) )
4745, 46eqeq12d 2455 . . 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 222 . 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 10577 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5014, 49sylib 196 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5135, 48, 50mpjaod 381 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   0cc0 9278    x. cmul 9283   NNcn 10318   NN0cn0 10575   Basecbs 14170   0gc0g 14374   Mndcmnd 15405  .gcmg 15410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-mnd 15411  df-mulg 15541
This theorem is referenced by:  mulgass  15650  odmodnn0  16036
  Copyright terms: Public domain W3C validator