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Theorem submod 14896
Description: The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
submod.h  |-  H  =  ( Gs  Y )
submod.o  |-  O  =  ( od `  G
)
submod.p  |-  P  =  ( od `  H
)
Assertion
Ref Expression
submod  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )

Proof of Theorem submod
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 730 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  Y  e.  (SubMnd `  G )
)
2 nnnn0 9988 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  NN0 )
32adantl 452 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  x  e.  NN0 )
4 simplr 731 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  A  e.  Y )
5 eqid 2296 . . . . . . 7  |-  (.g `  G
)  =  (.g `  G
)
6 submod.h . . . . . . 7  |-  H  =  ( Gs  Y )
7 eqid 2296 . . . . . . 7  |-  (.g `  H
)  =  (.g `  H
)
85, 6, 7submmulg 14618 . . . . . 6  |-  ( ( Y  e.  (SubMnd `  G )  /\  x  e.  NN0  /\  A  e.  Y )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
91, 3, 4, 8syl3anc 1182 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
10 eqid 2296 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
116, 10subm0 14449 . . . . . 6  |-  ( Y  e.  (SubMnd `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1211ad2antrr 706 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  ( 0g `  G )  =  ( 0g `  H
) )
139, 12eqeq12d 2310 . . . 4  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
( x (.g `  G
) A )  =  ( 0g `  G
)  <->  ( x (.g `  H ) A )  =  ( 0g `  H ) ) )
1413rabbidva 2792 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) } )
15 eqeq1 2302 . . . 4  |-  ( { x  e.  NN  | 
( x (.g `  G
) A )  =  ( 0g `  G
) }  =  {
x  e.  NN  | 
( x (.g `  H
) A )  =  ( 0g `  H
) }  ->  ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) }  =  (/)  <->  { x  e.  NN  | 
( x (.g `  H
) A )  =  ( 0g `  H
) }  =  (/) ) )
16 supeq1 7214 . . . 4  |-  ( { x  e.  NN  | 
( x (.g `  G
) A )  =  ( 0g `  G
) }  =  {
x  e.  NN  | 
( x (.g `  H
) A )  =  ( 0g `  H
) }  ->  sup ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) } ,  RR ,  `'  <  )  =  sup ( { x  e.  NN  | 
( x (.g `  H
) A )  =  ( 0g `  H
) } ,  RR ,  `'  <  ) )
1715, 16ifbieq2d 3598 . . 3  |-  ( { x  e.  NN  | 
( x (.g `  G
) A )  =  ( 0g `  G
) }  =  {
x  e.  NN  | 
( x (.g `  H
) A )  =  ( 0g `  H
) }  ->  if ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) }  =  (/)
,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) } ,  RR ,  `'  <  ) )  =  if ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g `  H ) }  =  (/)
,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g `  H ) } ,  RR ,  `'  <  ) ) )
1814, 17syl 15 . 2  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  if ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) }  =  (/)
,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g `  G ) } ,  RR ,  `'  <  ) )  =  if ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g `  H ) }  =  (/)
,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g `  H ) } ,  RR ,  `'  <  ) ) )
19 eqid 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2019submss 14443 . . . 4  |-  ( Y  e.  (SubMnd `  G
)  ->  Y  C_  ( Base `  G ) )
2120sselda 3193 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  G
) )
22 submod.o . . . 4  |-  O  =  ( od `  G
)
23 eqid 2296 . . . 4  |-  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }
2419, 5, 10, 22, 23odval 14865 . . 3  |-  ( A  e.  ( Base `  G
)  ->  ( O `  A )  =  if ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  (/) ,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) } ,  RR ,  `'  <  ) ) )
2521, 24syl 15 . 2  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  if ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  (/) ,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) } ,  RR ,  `'  <  ) ) )
26 simpr 447 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  Y )
2720adantr 451 . . . . 5  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  C_  ( Base `  G
) )
286, 19ressbas2 13215 . . . . 5  |-  ( Y 
C_  ( Base `  G
)  ->  Y  =  ( Base `  H )
)
2927, 28syl 15 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  =  ( Base `  H
) )
3026, 29eleqtrd 2372 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  H
) )
31 eqid 2296 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2296 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
33 submod.p . . . 4  |-  P  =  ( od `  H
)
34 eqid 2296 . . . 4  |-  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }  =  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }
3531, 7, 32, 33, 34odval 14865 . . 3  |-  ( A  e.  ( Base `  H
)  ->  ( P `  A )  =  if ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }  =  (/) ,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) } ,  RR ,  `'  <  ) ) )
3630, 35syl 15 . 2  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( P `  A )  =  if ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }  =  (/) ,  0 ,  sup ( { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) } ,  RR ,  `'  <  ) ) )
3718, 25, 363eqtr4d 2338 1  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   (/)c0 3468   ifcif 3578   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    < clt 8883   NNcn 9762   NN0cn0 9981   Basecbs 13164   ↾s cress 13165   0gc0g 13416  .gcmg 14382  SubMndcsubmnd 14430   odcod 14856
This theorem is referenced by:  subgod  14897
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-sup 7210  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  df-od 14860
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