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Theorem mulgnn0dir 15754
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 757 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  G  e.  Mnd )
2 simplr 754 . . . 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 996 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54ad2antrr 725 . . . 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 15753 . . . 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 1221 . . 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 753 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  G  e.  Mnd )
12 simpr1 994 . . . . . . . 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 1032 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  X  e.  B )
156, 7mulgnn0cl 15747 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  M  e.  NN0  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
1611, 13, 14, 15syl3anc 1219 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  .x.  X )  e.  B )
17 eqid 2451 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
186, 8, 17mndrid 15546 . . . . . 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 6207 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
226, 17, 7mulg0 15736 . . . . . . . 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 2492 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0g `  G
) )
2524oveq2d 6208 . . . . 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 6208 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  ( M  + 
0 ) )
2713nn0cnd 10741 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  CC )
2827addid1d 9672 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  0 )  =  M )
2926, 28eqtrd 2492 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  M )
3029oveq1d 6207 . . . . 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 2503 . . . 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 714 . . 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 995 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
34 elnn0 10684 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3533, 34sylib 196 . . . 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 784 . 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 753 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  G  e.  Mnd )
39 simplr2 1031 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  NN0 )
40 simplr3 1032 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  X  e.  B )
416, 7mulgnn0cl 15747 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
4238, 39, 40, 41syl3anc 1219 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( N  .x.  X )  e.  B )
436, 8, 17mndlid 15545 . . . 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 6207 . . . . 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 2492 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0g `  G
) )
4948oveq1d 6207 . . 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 6207 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  ( 0  +  N ) )
5139nn0cnd 10741 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  CC )
5251addid2d 9673 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  +  N )  =  N )
5350, 52eqtrd 2492 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  N )
5453oveq1d 6207 . . 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 2503 . 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 10684 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5712, 56sylib 196 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5837, 55, 57mpjaodan 784 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 965    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   0cc0 9385    + caddc 9388   NNcn 10425   NN0cn0 10682   Basecbs 14278   +g cplusg 14342   0gc0g 14482   Mndcmnd 15513  .gcmg 15518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-seq 11910  df-0g 14484  df-mnd 15519  df-mulg 15652
This theorem is referenced by:  mulgdirlem  15755  odmodnn0  16149  mndodconglem  16150  srgbinomlem  16750  evlslem1  17710  omndmul2  26311  omndmul3  26312  cpmadugsumlemB  31330
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