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Theorem mulgnn0ass 15971
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 1004 . . . . . . 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 15970 . . . . . 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 1230 . . . . 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 2467 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
126, 11, 7mulg0 15947 . . . . . . . 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 1002 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1514nn0cnd 10850 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  CC )
1615mul01d 9774 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  x.  0 )  =  0 )
1716oveq1d 6297 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( 0  .x.  X
) )
1813oveq2d 6298 . . . . . . . 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 15962 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  M  e.  NN0 )  -> 
( M  .x.  ( 0g `  G ) )  =  ( 0g `  G ) )
20193ad2antr1 1161 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0g `  G
) )  =  ( 0g `  G ) )
2118, 20eqtrd 2508 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( 0g `  G ) )
2213, 17, 213eqtr4d 2518 . . . . . 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 6290 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
2524oveq1d 6297 . . . . . 6  |-  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( M  x.  0 )  .x.  X ) )
26 oveq1 6289 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
2726oveq2d 6298 . . . . . 6  |-  ( N  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( M  .x.  (
0  .x.  X )
) )
2825, 27eqeq12d 2489 . . . . 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 1003 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
31 elnn0 10793 . . . . . 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 10850 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  CC )
3736mul02d 9773 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0  x.  N )  =  0 )
3837oveq1d 6297 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  X
) )
396, 7mulgnn0cl 15958 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
40393adant3r1 1205 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
416, 11, 7mulg0 15947 . . . . 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 2518 . . 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 6289 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
4544oveq1d 6297 . . . 4  |-  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( 0  x.  N )  .x.  X ) )
46 oveq1 6289 . . . 4  |-  ( M  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( 0  .x.  ( N  .x.  X ) ) )
4745, 46eqeq12d 2489 . . 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 10793 . . 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 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   0cc0 9488    x. cmul 9493   NNcn 10532   NN0cn0 10791   Basecbs 14486   0gc0g 14691   Mndcmnd 15722  .gcmg 15727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-0g 14693  df-mnd 15728  df-mulg 15861
This theorem is referenced by:  mulgass  15972  odmodnn0  16360
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