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Theorem submmulg 15750
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 991 . . . . . 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 2450 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 14368 . . . . . 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 11900 . . . 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 5777 . . 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 461 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  N  e.  NN )
9 eqid 2450 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
109submss 15566 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
11103ad2ant1 1009 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
12 simp3 990 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  S )
1311, 12sseldd 3441 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
1413adantr 465 . . . 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 2450 . . . . 5  |-  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
179, 3, 15, 16mulgnn 15721 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( N  .xb  X
)  =  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
188, 14, 17syl2anc 661 . . 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 15571 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
20193ad2ant1 1009 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
2112, 20eleqtrd 2538 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
2221adantr 465 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  X  e.  ( Base `  H
) )
23 eqid 2450 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
24 eqid 2450 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
25 submmulg.t . . . . 5  |-  .x.  =  (.g
`  H )
26 eqid 2450 . . . . 5  |-  seq 1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
2723, 24, 25, 26mulgnn 15721 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  ( Base `  H ) )  -> 
( N  .x.  X
)  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
288, 22, 27syl2anc 661 . . 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 2500 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .xb  X )  =  ( N  .x.  X
) )
30 simpl1 991 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  S  e.  (SubMnd `  G
) )
31 eqid 2450 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
322, 31subm0 15572 . . . . 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 465 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  G ) )
359, 31, 15mulg0 15720 . . . . 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 465 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  X  e.  ( Base `  H ) )
38 eqid 2450 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
3923, 38, 25mulg0 15720 . . . . 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 2500 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( 0  .xb  X
)  =  ( 0 
.x.  X ) )
42 simpr 461 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  N  =  0 )
4342oveq1d 6191 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( 0 
.xb  X ) )
4442oveq1d 6191 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  ( 0 
.x.  X ) )
4541, 43, 443eqtr4d 2500 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .xb  X
)  =  ( N 
.x.  X ) )
46 simp2 989 . . 3  |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  N  e.  NN0 )
47 elnn0 10668 . . 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 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    C_ wss 3412   {csn 3961    X. cxp 4922   ` cfv 5502  (class class class)co 6176   0cc0 9369   1c1 9370   NNcn 10409   NN0cn0 10666    seqcseq 11893   Basecbs 14262   ↾s cress 14263   +g cplusg 14326   0gc0g 14466  .gcmg 15502  SubMndcsubmnd 15551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-seq 11894  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-0g 14468  df-mnd 15503  df-submnd 15553  df-mulg 15636
This theorem is referenced by:  submod  16158  dchrfi  22696  dchrabs  22701  lgsqrlem1  22782  lgseisenlem4  22793  dchrisum0flblem1  22859  submarchi  26323  idomodle  29685  proot1ex  29693
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