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Theorem mgpress 17024
Description: Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypotheses
Ref Expression
mgpress.1  |-  S  =  ( Rs  A )
mgpress.2  |-  M  =  (mulGrp `  R )
Assertion
Ref Expression
mgpress  |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )

Proof of Theorem mgpress
StepHypRef Expression
1 mgpress.2 . . 3  |-  M  =  (mulGrp `  R )
2 simpr 461 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Base `  R )  C_  A
)
3 fvex 5882 . . . . . 6  |-  (mulGrp `  R )  e.  _V
41, 3eqeltri 2551 . . . . 5  |-  M  e. 
_V
54a1i 11 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  M  e.  _V )
6 simplr 754 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  A  e.  W )
7 eqid 2467 . . . . 5  |-  ( Ms  A )  =  ( Ms  A )
8 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
91, 8mgpbas 17019 . . . . 5  |-  ( Base `  R )  =  (
Base `  M )
107, 9ressid2 14560 . . . 4  |-  ( ( ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  M )
112, 5, 6, 10syl3anc 1228 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  M )
12 simpll 753 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  R  e.  V )
13 mgpress.1 . . . . . 6  |-  S  =  ( Rs  A )
1413, 8ressid2 14560 . . . . 5  |-  ( ( ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  R )
152, 12, 6, 14syl3anc 1228 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  S  =  R )
1615fveq2d 5876 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  (mulGrp `  S
)  =  (mulGrp `  R ) )
171, 11, 163eqtr4a 2534 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  (mulGrp `  S ) )
18 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18mgpval 17016 . . . 4  |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
2019oveq1i 6305 . . 3  |-  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )  =  ( ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R ) >. ) sSet  <.
( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. )
21 simpr 461 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  -.  ( Base `  R )  C_  A )
224a1i 11 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  M  e.  _V )
23 simplr 754 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  A  e.  W )
247, 9ressval2 14561 . . . 4  |-  ( ( -.  ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
2521, 22, 23, 24syl3anc 1228 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
26 eqid 2467 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
27 eqid 2467 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2826, 27mgpval 17016 . . . . 5  |-  (mulGrp `  S )  =  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )
29 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  R  e.  V )
3013, 8ressval2 14561 . . . . . . 7  |-  ( ( -.  ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  ( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
3121, 29, 23, 30syl3anc 1228 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  S  =  ( R sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
3213, 18ressmulr 14625 . . . . . . . . 9  |-  ( A  e.  W  ->  ( .r `  R )  =  ( .r `  S
) )
3332eqcomd 2475 . . . . . . . 8  |-  ( A  e.  W  ->  ( .r `  S )  =  ( .r `  R
) )
3433ad2antlr 726 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( .r `  S )  =  ( .r `  R
) )
3534opeq2d 4226 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  <. ( +g  `  ndx ) ,  ( .r `  S
) >.  =  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
3631, 35oveq12d 6313 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )  =  ( ( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. ) sSet  <.
( +g  `  ndx ) ,  ( .r `  R ) >. )
)
3728, 36syl5eq 2520 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  (mulGrp `  S )  =  ( ( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) sSet  <. ( +g  ` 
ndx ) ,  ( .r `  R )
>. ) )
38 1ne2 10760 . . . . . . 7  |-  1  =/=  2
3938necomi 2737 . . . . . 6  |-  2  =/=  1
40 plusgndx 14606 . . . . . . 7  |-  ( +g  ` 
ndx )  =  2
41 basendx 14557 . . . . . . 7  |-  ( Base `  ndx )  =  1
4240, 41neeq12i 2756 . . . . . 6  |-  ( ( +g  `  ndx )  =/=  ( Base `  ndx ) 
<->  2  =/=  1 )
4339, 42mpbir 209 . . . . 5  |-  ( +g  ` 
ndx )  =/=  ( Base `  ndx )
44 fvex 5882 . . . . . 6  |-  ( .r
`  R )  e. 
_V
45 fvex 5882 . . . . . . 7  |-  ( Base `  R )  e.  _V
4645inex2 4595 . . . . . 6  |-  ( A  i^i  ( Base `  R
) )  e.  _V
47 fvex 5882 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  _V
48 fvex 5882 . . . . . . 7  |-  ( Base `  ndx )  e.  _V
4947, 48setscom 14537 . . . . . 6  |-  ( ( ( R  e.  V  /\  ( +g  `  ndx )  =/=  ( Base `  ndx ) )  /\  (
( .r `  R
)  e.  _V  /\  ( A  i^i  ( Base `  R ) )  e.  _V ) )  ->  ( ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R ) >. ) sSet  <.
( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. )  =  (
( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) ) >.
) sSet  <. ( +g  `  ndx ) ,  ( .r `  R ) >. )
)
5044, 46, 49mpanr12 685 . . . . 5  |-  ( ( R  e.  V  /\  ( +g  `  ndx )  =/=  ( Base `  ndx ) )  ->  (
( R sSet  <. ( +g  ` 
ndx ) ,  ( .r `  R )
>. ) sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) ) >.
)  =  ( ( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. ) sSet  <.
( +g  `  ndx ) ,  ( .r `  R ) >. )
)
5129, 43, 50sylancl 662 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  (
( R sSet  <. ( +g  ` 
ndx ) ,  ( .r `  R )
>. ) sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) ) >.
)  =  ( ( R sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. ) sSet  <.
( +g  `  ndx ) ,  ( .r `  R ) >. )
)
5237, 51eqtr4d 2511 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  (mulGrp `  S )  =  ( ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. ) sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
5320, 25, 523eqtr4a 2534 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  (mulGrp `  S ) )
5417, 53pm2.61dan 789 1  |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    i^i cin 3480    C_ wss 3481   <.cop 4039   ` cfv 5594  (class class class)co 6295   1c1 9505   2c2 10597   ndxcnx 14504   sSet csts 14505   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   .rcmulr 14573  mulGrpcmgp 17013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-mgp 17014
This theorem is referenced by:  subrgcrng  17304  subrgsubm  17313  resrhm  17329  nn0srg  18356  rge0srg  18357  zringmpg  18391  m2cpmmhm  19115  rdivmuldivd  27606  xrge0iifmhm  27746  xrge0pluscn  27747  xrge0tmd  27753
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