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Theorem mulgnn0ass 12894
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 mndsgrp 12701 . . . . . . . 8  |-  ( G  e.  Mnd  ->  G  e. Smgrp )
21adantr 276 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  G  e. Smgrp )
32adantr 276 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  G  e. Smgrp )
4 simprl 529 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  M  e.  NN )
5 simprr 531 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  N  e.  NN )
6 simpr3 1005 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
76adantr 276 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  X  e.  B )
8 mulgass.b . . . . . . 7  |-  B  =  ( Base `  G
)
9 mulgass.t . . . . . . 7  |-  .x.  =  (.g
`  G )
108, 9mulgnnass 12893 . . . . . 6  |-  ( ( G  e. Smgrp  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B ) )  -> 
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) ) )
113, 4, 5, 7, 10syl13anc 1240 . . . . 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 ) ) )
1211expr 375 . . . 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 ) ) ) )
13 eqid 2177 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
148, 13, 9mulg0 12864 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
156, 14syl 14 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0 
.x.  X )  =  ( 0g `  G
) )
16 simpr1 1003 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1716nn0cnd 9207 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  CC )
1817mul01d 8327 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  x.  0 )  =  0 )
1918oveq1d 5883 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( 0  .x.  X
) )
2015oveq2d 5884 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( M  .x.  ( 0g
`  G ) ) )
218, 9, 13mulgnn0z 12885 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  M  e.  NN0 )  -> 
( M  .x.  ( 0g `  G ) )  =  ( 0g `  G ) )
22213ad2antr1 1162 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0g `  G
) )  =  ( 0g `  G ) )
2320, 22eqtrd 2210 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  .x.  ( 0  .x.  X
) )  =  ( 0g `  G ) )
2415, 19, 233eqtr4d 2220 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) )
2524adantr 276 . . . . 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 )
) )
26 oveq2 5876 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
2726oveq1d 5883 . . . . . 6  |-  ( N  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( M  x.  0 )  .x.  X ) )
28 oveq1 5875 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
2928oveq2d 5884 . . . . . 6  |-  ( N  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( M  .x.  (
0  .x.  X )
) )
3027, 29eqeq12d 2192 . . . . 5  |-  ( N  =  0  ->  (
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) )  <->  ( ( M  x.  0 ) 
.x.  X )  =  ( M  .x.  (
0  .x.  X )
) ) )
3125, 30syl5ibrcom 157 . . . 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 ) ) ) )
32 simpr2 1004 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
33 elnn0 9154 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3432, 33sylib 122 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3534adantr 276 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3612, 31, 35mpjaod 718 . . 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 ) ) )
3736ex 115 . 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 ) ) ) )
3832nn0cnd 9207 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  CC )
3938mul02d 8326 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0  x.  N )  =  0 )
4039oveq1d 5883 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  X
) )
418, 9mulgnn0cl 12875 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
42413adant3r1 1212 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
438, 13, 9mulg0 12864 . . . . 5  |-  ( ( N  .x.  X )  e.  B  ->  (
0  .x.  ( N  .x.  X ) )  =  ( 0g `  G
) )
4442, 43syl 14 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( 0 
.x.  ( N  .x.  X ) )  =  ( 0g `  G
) )
4515, 40, 443eqtr4d 2220 . . 3  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  ( N  .x.  X ) ) )
46 oveq1 5875 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
4746oveq1d 5883 . . . 4  |-  ( M  =  0  ->  (
( M  x.  N
)  .x.  X )  =  ( ( 0  x.  N )  .x.  X ) )
48 oveq1 5875 . . . 4  |-  ( M  =  0  ->  ( M  .x.  ( N  .x.  X ) )  =  ( 0  .x.  ( N  .x.  X ) ) )
4947, 48eqeq12d 2192 . . 3  |-  ( M  =  0  ->  (
( ( M  x.  N )  .x.  X
)  =  ( M 
.x.  ( N  .x.  X ) )  <->  ( (
0  x.  N ) 
.x.  X )  =  ( 0  .x.  ( N  .x.  X ) ) ) )
5045, 49syl5ibrcom 157 . 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 ) ) ) )
51 elnn0 9154 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5216, 51sylib 122 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5337, 50, 52mpjaod 718 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    /\ wa 104    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868   0cc0 7789    x. cmul 7794   NNcn 8895   NN0cn0 9152   Basecbs 12432   0gc0g 12640  Smgrpcsgrp 12686   Mndcmnd 12696  .gcmg 12859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-fz 9983  df-fzo 10116  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-minusg 12758  df-mulg 12860
This theorem is referenced by:  mulgass  12895
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