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Theorem mulgnn0dir 14840
Description: Sum of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgnndir.b  |-  B  =  ( Base `  G
)
mulgnndir.t  |-  .x.  =  (.g
`  G )
mulgnndir.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
mulgnn0dir  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X
)  .+  ( N  .x.  X ) ) )

Proof of Theorem mulgnn0dir
StepHypRef Expression
1 simplll 735 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  G  e.  Mnd )
2 simplr 732 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  M  e.  NN )
3 simpr 448 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  N  e.  NN )
4 simpr3 965 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54ad2antrr 707 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  X  e.  B
)
6 mulgnndir.b . . . . 5  |-  B  =  ( Base `  G
)
7 mulgnndir.t . . . . 5  |-  .x.  =  (.g
`  G )
8 mulgnndir.p . . . . 5  |-  .+  =  ( +g  `  G )
96, 7, 8mulgnndir 14839 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B )
)  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X
)  .+  ( N  .x.  X ) ) )
101, 2, 3, 5, 9syl13anc 1186 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X
)  .+  ( N  .x.  X ) ) )
11 simpll 731 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  G  e.  Mnd )
12 simpr1 963 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1312adantr 452 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  NN0 )
14 simplr3 1001 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  X  e.  B )
156, 7mulgnn0cl 14833 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  M  e.  NN0  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
1611, 13, 14, 15syl3anc 1184 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  .x.  X )  e.  B )
17 eqid 2387 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
186, 8, 17mndrid 14644 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .+  ( 0g `  G ) )  =  ( M  .x.  X ) )
1911, 16, 18syl2anc 643 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
( M  .x.  X
)  .+  ( 0g `  G ) )  =  ( M  .x.  X
) )
20 simpr 448 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  N  =  0 )
2120oveq1d 6035 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
226, 17, 7mulg0 14822 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2314, 22syl 16 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2421, 23eqtrd 2419 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0g `  G
) )
2524oveq2d 6036 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
( M  .x.  X
)  .+  ( N  .x.  X ) )  =  ( ( M  .x.  X )  .+  ( 0g `  G ) ) )
2620oveq2d 6036 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  ( M  + 
0 ) )
2713nn0cnd 10208 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  CC )
2827addid1d 9198 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  0 )  =  M )
2926, 28eqtrd 2419 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  M )
3029oveq1d 6035 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
( M  +  N
)  .x.  X )  =  ( M  .x.  X ) )
3119, 25, 303eqtr4rd 2430 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
( M  +  N
)  .x.  X )  =  ( ( M 
.x.  X )  .+  ( N  .x.  X ) ) )
3231adantlr 696 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  =  0
)  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X
)  .+  ( N  .x.  X ) ) )
33 simpr2 964 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
34 elnn0 10155 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3533, 34sylib 189 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3635adantr 452 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3710, 32, 36mpjaodan 762 . 2  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( ( M  +  N ) 
.x.  X )  =  ( ( M  .x.  X )  .+  ( N  .x.  X ) ) )
38 simpll 731 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  G  e.  Mnd )
39 simplr2 1000 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  NN0 )
40 simplr3 1001 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  X  e.  B )
416, 7mulgnn0cl 14833 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
4238, 39, 40, 41syl3anc 1184 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( N  .x.  X )  e.  B )
436, 8, 17mndlid 14643 . . . 4  |-  ( ( G  e.  Mnd  /\  ( N  .x.  X )  e.  B )  -> 
( ( 0g `  G )  .+  ( N  .x.  X ) )  =  ( N  .x.  X ) )
4438, 42, 43syl2anc 643 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
( 0g `  G
)  .+  ( N  .x.  X ) )  =  ( N  .x.  X
) )
45 simpr 448 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  M  =  0 )
4645oveq1d 6035 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0  .x.  X
) )
4740, 22syl 16 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
4846, 47eqtrd 2419 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0g `  G
) )
4948oveq1d 6035 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
( M  .x.  X
)  .+  ( N  .x.  X ) )  =  ( ( 0g `  G )  .+  ( N  .x.  X ) ) )
5045oveq1d 6035 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  ( 0  +  N ) )
5139nn0cnd 10208 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  CC )
5251addid2d 9199 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  +  N )  =  N )
5350, 52eqtrd 2419 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  N )
5453oveq1d 6035 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
( M  +  N
)  .x.  X )  =  ( N  .x.  X ) )
5544, 49, 543eqtr4rd 2430 . 2  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
( M  +  N
)  .x.  X )  =  ( ( M 
.x.  X )  .+  ( N  .x.  X ) ) )
56 elnn0 10155 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5712, 56sylib 189 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5837, 55, 57mpjaodan 762 1  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X
)  .+  ( N  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   0cc0 8923    + caddc 8926   NNcn 9932   NN0cn0 10153   Basecbs 13396   +g cplusg 13456   0gc0g 13650   Mndcmnd 14611  .gcmg 14616
This theorem is referenced by:  mulgdirlem  14841  odmodnn0  15105  mndodconglem  15106  evlslem1  19803
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-seq 11251  df-0g 13654  df-mnd 14617  df-mulg 14742
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