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Theorem submmulg 14927
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 961 . . . . . 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 2438 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13573 . . . . . 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 11332 . . . 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 5732 . . 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 449 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  N  e.  NN )
9 eqid 2438 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
109submss 14752 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
11103ad2ant1 979 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
12 simp3 960 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  S )
1311, 12sseldd 3351 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
1413adantr 453 . . . 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 2438 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
179, 3, 15, 16mulgnn 14898 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( N  .xb  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
188, 14, 17syl2anc 644 . . 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 14757 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
20193ad2ant1 979 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
2112, 20eleqtrd 2514 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
2221adantr 453 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  X  e.  ( Base `  H
) )
23 eqid 2438 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
24 eqid 2438 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
25 submmulg.t . . . . 5  |-  .x.  =  (.g
`  H )
26 eqid 2438 . . . . 5  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
2723, 24, 25, 26mulgnn 14898 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  H ) )  -> 
( N  .x.  X
)  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
288, 22, 27syl2anc 644 . . 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 2480 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )
30 simpl1 961 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  S  e.  (SubMnd `  G
) )
31 eqid 2438 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
322, 31subm0 14758 . . . . 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 453 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  G ) )
359, 31, 15mulg0 14897 . . . . 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 453 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  H ) )
38 eqid 2438 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
3923, 38, 25mulg0 14897 . . . . 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 2480 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0 
.x.  X ) )
42 simpr 449 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  N  =  0 )
4342oveq1d 6098 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( 0 
.xb  X ) )
4442oveq1d 6098 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  ( 0 
.x.  X ) )
4541, 43, 443eqtr4d 2480 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( N 
.x.  X ) )
46 simp2 959 . . 3  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  N  e.  NN0 )
47 elnn0 10225 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
4846, 47sylib 190 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  e.  NN  \/  N  =  0 ) )
4929, 45, 48mpjaodan 763 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 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   {csn 3816    X. cxp 4878   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993   NNcn 10002   NN0cn0 10223    seq cseq 11325   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   0gc0g 13725  .gcmg 14691  SubMndcsubmnd 14739
This theorem is referenced by:  submod  15205  dchrfi  21041  dchrabs  21046  lgsqrlem1  21127  lgseisenlem4  21138  dchrisum0flblem1  21204  idomodle  27491  proot1ex  27499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-seq 11326  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-mulg 14817
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