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Theorem submmulg 14602
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 2283 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13250 . . . . . 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 11053 . . . 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 5527 . . 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 2283 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
109submss 14427 . . . . . . 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 3181 . . . . 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 2283 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
179, 3, 15, 16mulgnn 14573 . . . 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 14432 . . . . . . 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 2359 . . . . 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 2283 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
24 eqid 2283 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
25 submmulg.t . . . . 5  |-  .x.  =  (.g
`  H )
26 eqid 2283 . . . . 5  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
2723, 24, 25, 26mulgnn 14573 . . . 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 2325 . 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 2283 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
322, 31subm0 14433 . . . . 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 14572 . . . . 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 2283 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
3923, 38, 25mulg0 14572 . . . . 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 2325 . . 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 5873 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( 0 
.xb  X ) )
4442oveq1d 5873 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  ( 0 
.x.  X ) )
4541, 43, 443eqtr4d 2325 . 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 9967 . . 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 1623    e. wcel 1684    C_ wss 3152   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   NNcn 9746   NN0cn0 9965    seq cseq 11046   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400  .gcmg 14366  SubMndcsubmnd 14414
This theorem is referenced by:  submod  14880  dchrfi  20494  dchrabs  20499  lgsqrlem1  20580  lgseisenlem4  20591  dchrisum0flblem1  20657  idomodle  27512  proot1ex  27520
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-2 9804  df-n0 9966  df-z 10025  df-seq 11047  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-mulg 14492
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