MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgnn0dir Unicode version

Theorem mulgnn0dir 14590
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 734 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  G  e.  Mnd )
2 simplr 731 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  M  e.  NN )
3 simpr 447 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  ( M  e.  NN0  /\  N  e. 
NN0  /\  X  e.  B ) )  /\  M  e.  NN )  /\  N  e.  NN )  ->  N  e.  NN )
4 simpr3 963 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  X  e.  B )
54ad2antrr 706 . . . 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 14589 . . . 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 1184 . . 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 730 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  G  e.  Mnd )
12 simpr1 961 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  M  e.  NN0 )
1312adantr 451 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  NN0 )
14 simplr3 999 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  X  e.  B )
156, 7mulgnn0cl 14583 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  M  e.  NN0  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
1611, 13, 14, 15syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  .x.  X )  e.  B )
17 eqid 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
186, 8, 17mndrid 14394 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( M  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .+  ( 0g `  G ) )  =  ( M  .x.  X ) )
1911, 16, 18syl2anc 642 . . . . 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 447 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  N  =  0 )
2120oveq1d 5873 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
226, 17, 7mulg0 14572 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2314, 22syl 15 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2421, 23eqtrd 2315 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0g `  G
) )
2524oveq2d 5874 . . . . 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 5874 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  ( M  + 
0 ) )
2713nn0cnd 10020 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  CC )
2827addid1d 9012 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  0 )  =  M )
2926, 28eqtrd 2315 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  M )
3029oveq1d 5873 . . . . 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 2326 . . . 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 695 . . 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 962 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  N  e.  NN0 )
34 elnn0 9967 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3533, 34sylib 188 . . . 4  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( N  e.  NN  \/  N  =  0 ) )
3635adantr 451 . . 3  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  e.  NN )  ->  ( N  e.  NN  \/  N  =  0 ) )
3710, 32, 36mpjaodan 761 . 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 730 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  G  e.  Mnd )
39 simplr2 998 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  NN0 )
40 simplr3 999 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  X  e.  B )
416, 7mulgnn0cl 14583 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
4238, 39, 40, 41syl3anc 1182 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( N  .x.  X )  e.  B )
436, 8, 17mndlid 14393 . . . 4  |-  ( ( G  e.  Mnd  /\  ( N  .x.  X )  e.  B )  -> 
( ( 0g `  G )  .+  ( N  .x.  X ) )  =  ( N  .x.  X ) )
4438, 42, 43syl2anc 642 . . 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 447 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  M  =  0 )
4645oveq1d 5873 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0  .x.  X
) )
4740, 22syl 15 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
4846, 47eqtrd 2315 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0g `  G
) )
4948oveq1d 5873 . . 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 5873 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  ( 0  +  N ) )
5139nn0cnd 10020 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  CC )
5251addid2d 9013 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  +  N )  =  N )
5350, 52eqtrd 2315 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  N )
5453oveq1d 5873 . . 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 2326 . 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 9967 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
5712, 56sylib 188 . 2  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  ->  ( M  e.  NN  \/  M  =  0 ) )
5837, 55, 57mpjaodan 761 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 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   0cc0 8737    + caddc 8740   NNcn 9746   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361  .gcmg 14366
This theorem is referenced by:  mulgdirlem  14591  odmodnn0  14855  mndodconglem  14856  evlslem1  19399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-mulg 14492
  Copyright terms: Public domain W3C validator