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Theorem mulgnn0dir 14606
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 14605 . . . 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 14599 . . . . . . 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 2296 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
186, 8, 17mndrid 14410 . . . . . 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 5889 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
226, 17, 7mulg0 14588 . . . . . . . 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 2328 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( N  .x.  X )  =  ( 0g `  G
) )
2524oveq2d 5890 . . . . 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 5890 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  ( M  + 
0 ) )
2713nn0cnd 10036 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  M  e.  CC )
2827addid1d 9028 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  0 )  =  M )
2926, 28eqtrd 2328 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  N  = 
0 )  ->  ( M  +  N )  =  M )
3029oveq1d 5889 . . . . 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 2339 . . . 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 9983 . . . . 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 14599 . . . . 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 14409 . . . 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 5889 . . . . 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 2328 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  .x.  X )  =  ( 0g `  G
) )
4948oveq1d 5889 . . 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 5889 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  ( 0  +  N ) )
5139nn0cnd 10036 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  N  e.  CC )
5251addid2d 9029 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  (
0  +  N )  =  N )
5350, 52eqtrd 2328 . . . 4  |-  ( ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B )
)  /\  M  = 
0 )  ->  ( M  +  N )  =  N )
5453oveq1d 5889 . . 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 2339 . 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 9983 . . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   0cc0 8753    + caddc 8756   NNcn 9762   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377  .gcmg 14382
This theorem is referenced by:  mulgdirlem  14607  odmodnn0  14871  mndodconglem  14872  evlslem1  19415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-0g 13420  df-mnd 14383  df-mulg 14508
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