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Theorem submod 15191
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 731 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  Y  e.  (SubMnd `  G )
)
2 nnnn0 10217 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  NN0 )
32adantl 453 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  x  e.  NN0 )
4 simplr 732 . . . . . 6  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  A  e.  Y )
5 eqid 2435 . . . . . . 7  |-  (.g `  G
)  =  (.g `  G
)
6 submod.h . . . . . . 7  |-  H  =  ( Gs  Y )
7 eqid 2435 . . . . . . 7  |-  (.g `  H
)  =  (.g `  H
)
85, 6, 7submmulg 14913 . . . . . 6  |-  ( ( Y  e.  (SubMnd `  G )  /\  x  e.  NN0  /\  A  e.  Y )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
91, 3, 4, 8syl3anc 1184 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
x (.g `  G ) A )  =  ( x (.g `  H ) A ) )
10 eqid 2435 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
116, 10subm0 14744 . . . . . 6  |-  ( Y  e.  (SubMnd `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1211ad2antrr 707 . . . . 5  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  ( 0g `  G )  =  ( 0g `  H
) )
139, 12eqeq12d 2449 . . . 4  |-  ( ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  /\  x  e.  NN )  ->  (
( x (.g `  G
) A )  =  ( 0g `  G
)  <->  ( x (.g `  H ) A )  =  ( 0g `  H ) ) )
1413rabbidva 2939 . . 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 2441 . . . 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 7441 . . . 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 3751 . . 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 2435 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2019submss 14738 . . . 4  |-  ( Y  e.  (SubMnd `  G
)  ->  Y  C_  ( Base `  G ) )
2120sselda 3340 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  G
) )
22 submod.o . . . 4  |-  O  =  ( od `  G
)
23 eqid 2435 . . . 4  |-  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }  =  { x  e.  NN  |  ( x (.g `  G ) A )  =  ( 0g
`  G ) }
2419, 5, 10, 22, 23odval 15160 . . 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 448 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  Y )
2720adantr 452 . . . . 5  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  C_  ( Base `  G
) )
286, 19ressbas2 13508 . . . . 5  |-  ( Y 
C_  ( Base `  G
)  ->  Y  =  ( Base `  H )
)
2927, 28syl 16 . . . 4  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  Y  =  ( Base `  H
) )
3026, 29eleqtrd 2511 . . 3  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  A  e.  ( Base `  H
) )
31 eqid 2435 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2435 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
33 submod.p . . . 4  |-  P  =  ( od `  H
)
34 eqid 2435 . . . 4  |-  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }  =  { x  e.  NN  |  ( x (.g `  H ) A )  =  ( 0g
`  H ) }
3531, 7, 32, 33, 34odval 15160 . . 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 2477 1  |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   (/)c0 3620   ifcif 3731   `'ccnv 4868   ` cfv 5445  (class class class)co 6072   supcsup 7436   RRcr 8978   0cc0 8979    < clt 9109   NNcn 9989   NN0cn0 10210   Basecbs 13457   ↾s cress 13458   0gc0g 13711  .gcmg 14677  SubMndcsubmnd 14725   odcod 15151
This theorem is referenced by:  subgod  15192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-seq 11312  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-0g 13715  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-od 15155
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