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Theorem submod 16190
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 753 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  Y  e.  (SubMnd `  G )
)
2 nnnn0 10698 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  NN0 )
32adantl 466 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  x  e.  NN0 )
4 simplr 754 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  A  e.  Y )
5 eqid 2454 . . . . . . 7  |-  (.g `  G
)  =  (.g `  G
)
6 submod.h . . . . . . 7  |-  H  =  ( Gs  Y )
7 eqid 2454 . . . . . . 7  |-  (.g `  H
)  =  (.g `  H
)
85, 6, 7submmulg 15782 . . . . . 6  |-  ( ( Y  e.  (SubMnd `  G )  /\  x  e.  NN0  /\  A  e.  Y )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
91, 3, 4, 8syl3anc 1219 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
10 eqid 2454 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
116, 10subm0 15604 . . . . . 6  |-  ( Y  e.  (SubMnd `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1211ad2antrr 725 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  ( 0g `  G )  =  ( 0g `  H
) )
139, 12eqeq12d 2476 . . . 4  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
( x (.g `  G
) A )  =  ( 0g `  G
)  <->  ( x (.g `  H ) A )  =  ( 0g `  H ) ) )
1413rabbidva 3069 . . 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 2458 . . . 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 7807 . . . 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 3923 . . 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 16 . 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 2454 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2019submss 15598 . . . 4  |-  ( Y  e.  (SubMnd `  G
)  ->  Y  C_  ( Base `  G ) )
2120sselda 3465 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  G
) )
22 submod.o . . . 4  |-  O  =  ( od `  G
)
23 eqid 2454 . . . 4  |-  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }
2419, 5, 10, 22, 23odval 16159 . . 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 16 . 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 461 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  Y )
2720adantr 465 . . . . 5  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  C_  ( Base `  G
) )
286, 19ressbas2 14349 . . . . 5  |-  ( Y 
C_  ( Base `  G
)  ->  Y  =  ( Base `  H )
)
2927, 28syl 16 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  =  ( Base `  H
) )
3026, 29eleqtrd 2544 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  H
) )
31 eqid 2454 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2454 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
33 submod.p . . . 4  |-  P  =  ( od `  H
)
34 eqid 2454 . . . 4  |-  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }  =  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }
3531, 7, 32, 33, 34odval 16159 . . 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 16 . 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 2505 1  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3437   (/)c0 3746   ifcif 3900   `'ccnv 4948   ` cfv 5527  (class class class)co 6201   supcsup 7802   RRcr 9393   0cc0 9394    < clt 9530   NNcn 10434   NN0cn0 10691   Basecbs 14293   ↾s cress 14294   0gc0g 14498  .gcmg 15534  SubMndcsubmnd 15583   odcod 16150
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-seq 11925  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-mnd 15535  df-submnd 15585  df-mulg 15668  df-od 16154
This theorem is referenced by:  subgod  16191
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