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Theorem submmulg 16294
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 997 . . . . . 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 2382 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 14748 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
51, 4syl 16 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( +g  `  G )  =  ( +g  `  H
) )
65seqeq2d 12017 . . . 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 5776 . . 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 459 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  N  e.  NN )
9 eqid 2382 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
109submss 16098 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
11103ad2ant1 1015 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
12 simp3 996 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  S )
1311, 12sseldd 3418 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
1413adantr 463 . . . 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 2382 . . . . 5  |-  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
179, 3, 15, 16mulgnn 16265 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( N  .xb  X
)  =  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
188, 14, 17syl2anc 659 . . 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 16103 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
20193ad2ant1 1015 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
2112, 20eleqtrd 2472 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
2221adantr 463 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  X  e.  ( Base `  H
) )
23 eqid 2382 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
24 eqid 2382 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
25 submmulg.t . . . . 5  |-  .x.  =  (.g
`  H )
26 eqid 2382 . . . . 5  |-  seq 1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
2723, 24, 25, 26mulgnn 16265 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  H ) )  -> 
( N  .x.  X
)  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
288, 22, 27syl2anc 659 . . 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 2433 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )
30 simpl1 997 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  S  e.  (SubMnd `  G
) )
31 eqid 2382 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
322, 31subm0 16104 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3330, 32syl 16 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0g `  G
)  =  ( 0g
`  H ) )
3413adantr 463 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  G ) )
359, 31, 15mulg0 16264 . . . . 5  |-  ( X  e.  ( Base `  G
)  ->  ( 0 
.xb  X )  =  ( 0g `  G
) )
3634, 35syl 16 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0g
`  G ) )
3721adantr 463 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  H ) )
38 eqid 2382 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
3923, 38, 25mulg0 16264 . . . . 5  |-  ( X  e.  ( Base `  H
)  ->  ( 0 
.x.  X )  =  ( 0g `  H
) )
4037, 39syl 16 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .x.  X
)  =  ( 0g
`  H ) )
4133, 36, 403eqtr4d 2433 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0 
.x.  X ) )
42 simpr 459 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  N  =  0 )
4342oveq1d 6211 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( 0 
.xb  X ) )
4442oveq1d 6211 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  ( 0 
.x.  X ) )
4541, 43, 443eqtr4d 2433 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( N 
.x.  X ) )
46 simp2 995 . . 3  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  N  e.  NN0 )
47 elnn0 10714 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
4846, 47sylib 196 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  e.  NN  \/  N  =  0 ) )
4929, 45, 48mpjaodan 784 1  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   {csn 3944    X. cxp 4911   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404   NNcn 10452   NN0cn0 10712    seqcseq 12010   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   0gc0g 14847  SubMndcsubmnd 16082  .gcmg 16173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-seq 12011  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177
This theorem is referenced by:  submod  16706  dchrfi  23647  dchrabs  23652  lgsqrlem1  23733  lgseisenlem4  23744  dchrisum0flblem1  23810  submarchi  27883  idomodle  31321  proot1ex  31329
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