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Theorem mulgnn0dir 16491
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 762 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  G  e.  Mnd )
2 simplr 756 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  M  e.  NN )
3 simpr 461 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  N  e.  NN )
4 simpr3 1007 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54ad2antrr 726 . . . 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 16490 . . . 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 1234 . . 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 754 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  G  e.  Mnd )
12 simpr1 1005 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1312adantr 465 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  NN0 )
14 simplr3 1043 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  X  e.  B )
156, 7mulgnn0cl 16484 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  M  e.  NN0  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
1611, 13, 14, 15syl3anc 1232 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  .x.  X )  e.  B )
17 eqid 2404 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
186, 8, 17mndrid 16268 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .+  ( 0g `  G ) )  =  ( M  .x.  X ) )
1911, 16, 18syl2anc 661 . . . . 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 461 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  N  =  0 )
2120oveq1d 6295 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
226, 17, 7mulg0 16473 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2314, 22syl 17 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2421, 23eqtrd 2445 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0g `  G
) )
2524oveq2d 6296 . . . . 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 6296 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  ( M  + 
0 ) )
2713nn0cnd 10897 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  CC )
2827addid1d 9816 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  0 )  =  M )
2926, 28eqtrd 2445 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  M )
3029oveq1d 6295 . . . . 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 2456 . . . 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 715 . . 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 1006 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
34 elnn0 10840 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3533, 34sylib 198 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3635adantr 465 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3710, 32, 36mpjaodan 789 . 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 754 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  G  e.  Mnd )
39 simplr2 1042 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  NN0 )
40 simplr3 1043 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  X  e.  B )
416, 7mulgnn0cl 16484 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
4238, 39, 40, 41syl3anc 1232 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( N  .x.  X )  e.  B )
436, 8, 17mndlid 16267 . . . 4  |-  ( ( G  e.  Mnd  /\  ( N  .x.  X )  e.  B )  -> 
( ( 0g `  G )  .+  ( N  .x.  X ) )  =  ( N  .x.  X ) )
4438, 42, 43syl2anc 661 . . 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 461 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  M  =  0 )
4645oveq1d 6295 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0  .x.  X
) )
4740, 22syl 17 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
4846, 47eqtrd 2445 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0g `  G
) )
4948oveq1d 6295 . . 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 6295 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  ( 0  +  N ) )
5139nn0cnd 10897 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  CC )
5251addid2d 9817 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  +  N )  =  N )
5350, 52eqtrd 2445 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  N )
5453oveq1d 6295 . . 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 2456 . 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 10840 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5712, 56sylib 198 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5837, 55, 57mpjaodan 789 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   ` cfv 5571  (class class class)co 6280   0cc0 9524    + caddc 9527   NNcn 10578   NN0cn0 10838   Basecbs 14843   +g cplusg 14911   0gc0g 15056   Mndcmnd 16245  .gcmg 16382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-seq 12154  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mulg 16386
This theorem is referenced by:  mulgdirlem  16492  odmodnn0  16890  mndodconglem  16891  srgbinomlem  17517  evlslem1  18506  cpmadugsumlemB  19669  omndmul2  28167  omndmul3  28168
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