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Theorem mulgnn0ass 15661
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 753 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  G  e.  Mnd )
2 simprl 755 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  M  e.  NN )
3 simprr 756 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  N  e.  NN )
4 simpr3 996 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54adantr 465 . . . . . 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 15660 . . . . . 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 1220 . . . . 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 615 . . . 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 2443 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
126, 11, 7mulg0 15637 . . . . . . . 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 994 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1514nn0cnd 10643 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  CC )
1615mul01d 9573 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  x.  0 )  =  0 )
1716oveq1d 6111 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( 0  .x.  X
) )
1813oveq2d 6112 . . . . . . . 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 15652 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  M  e.  NN0 )  -> 
( M  .x.  ( 0g `  G ) )  =  ( 0g `  G ) )
20193ad2antr1 1153 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0g `  G
) )  =  ( 0g `  G ) )
2118, 20eqtrd 2475 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( 0g `  G ) )
2213, 17, 213eqtr4d 2485 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) )
2322adantr 465 . . . . 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 6104 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
2524oveq1d 6111 . . . . . 6  |-  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( M  x.  0 )  .x.  X ) )
26 oveq1 6103 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
2726oveq2d 6112 . . . . . 6  |-  ( N  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( M  .x.  (
0  .x.  X )
) )
2825, 27eqeq12d 2457 . . . . 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 995 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
31 elnn0 10586 . . . . . 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 465 . . . 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 10643 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  CC )
3736mul02d 9572 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0  x.  N )  =  0 )
3837oveq1d 6111 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  X
) )
396, 7mulgnn0cl 15648 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
40393adant3r1 1196 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
416, 11, 7mulg0 15637 . . . . 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 2485 . . 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 6103 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
4544oveq1d 6111 . . . 4  |-  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( 0  x.  N )  .x.  X ) )
46 oveq1 6103 . . . 4  |-  ( M  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( 0  .x.  ( N  .x.  X ) ) )
4745, 46eqeq12d 2457 . . 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 10586 . . 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 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   0cc0 9287    x. cmul 9292   NNcn 10327   NN0cn0 10584   Basecbs 14179   0gc0g 14383   Mndcmnd 15414  .gcmg 15419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-0g 14385  df-mnd 15420  df-mulg 15553
This theorem is referenced by:  mulgass  15662  odmodnn0  16048
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