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Theorem submmulg 14618
Description: A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
submmulgcl.t  |-  .xb  =  (.g
`  G )
submmulg.h  |-  H  =  ( Gs  S )
submmulg.t  |-  .x.  =  (.g
`  H )
Assertion
Ref Expression
submmulg  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )

Proof of Theorem submmulg
StepHypRef Expression
1 simpl1 958 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  S  e.  (SubMnd `  G )
)
2 submmulg.h . . . . . . 7  |-  H  =  ( Gs  S )
3 eqid 2296 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13266 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
51, 4syl 15 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( +g  `  G )  =  ( +g  `  H
) )
65seqeq2d 11069 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
76fveq1d 5543 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N )  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) )
8 simpr 447 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  N  e.  NN )
9 eqid 2296 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
109submss 14443 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
11103ad2ant1 976 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
12 simp3 957 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  S )
1311, 12sseldd 3194 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
1413adantr 451 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  X  e.  ( Base `  G
) )
15 submmulgcl.t . . . . 5  |-  .xb  =  (.g
`  G )
16 eqid 2296 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
179, 3, 15, 16mulgnn 14589 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( N  .xb  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
188, 14, 17syl2anc 642 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .xb  X )  =  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) )
192submbas 14448 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
20193ad2ant1 976 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
2112, 20eleqtrd 2372 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
2221adantr 451 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  X  e.  ( Base `  H
) )
23 eqid 2296 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
24 eqid 2296 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
25 submmulg.t . . . . 5  |-  .x.  =  (.g
`  H )
26 eqid 2296 . . . . 5  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
2723, 24, 25, 26mulgnn 14589 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  H ) )  -> 
( N  .x.  X
)  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
288, 22, 27syl2anc 642 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .x.  X )  =  (  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { X }
) ) `  N
) )
297, 18, 283eqtr4d 2338 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )
30 simpl1 958 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  S  e.  (SubMnd `  G
) )
31 eqid 2296 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
322, 31subm0 14449 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3330, 32syl 15 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0g `  G
)  =  ( 0g
`  H ) )
3413adantr 451 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  G ) )
359, 31, 15mulg0 14588 . . . . 5  |-  ( X  e.  ( Base `  G
)  ->  ( 0 
.xb  X )  =  ( 0g `  G
) )
3634, 35syl 15 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0g
`  G ) )
3721adantr 451 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  H ) )
38 eqid 2296 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
3923, 38, 25mulg0 14588 . . . . 5  |-  ( X  e.  ( Base `  H
)  ->  ( 0 
.x.  X )  =  ( 0g `  H
) )
4037, 39syl 15 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .x.  X
)  =  ( 0g
`  H ) )
4133, 36, 403eqtr4d 2338 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0 
.x.  X ) )
42 simpr 447 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  N  =  0 )
4342oveq1d 5889 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( 0 
.xb  X ) )
4442oveq1d 5889 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  ( 0 
.x.  X ) )
4541, 43, 443eqtr4d 2338 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( N 
.x.  X ) )
46 simp2 956 . . 3  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  N  e.  NN0 )
47 elnn0 9983 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
4846, 47sylib 188 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  e.  NN  \/  N  =  0 ) )
4929, 45, 48mpjaodan 761 1  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .xb  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    C_ wss 3165   {csn 3653    X. cxp 4703   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754   NNcn 9762   NN0cn0 9981    seq cseq 11062   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   0gc0g 13416  .gcmg 14382  SubMndcsubmnd 14430
This theorem is referenced by:  submod  14896  dchrfi  20510  dchrabs  20515  lgsqrlem1  20596  lgseisenlem4  20607  dchrisum0flblem1  20673  idomodle  27615  proot1ex  27623
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-2 9820  df-n0 9982  df-z 10041  df-seq 11063  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-mulg 14508
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