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Theorem mgpress 15352
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 447 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Base `  R )  C_  A
)
3 fvex 5555 . . . . . 6  |-  (mulGrp `  R )  e.  _V
41, 3eqeltri 2366 . . . . 5  |-  M  e. 
_V
54a1i 10 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  M  e.  _V )
6 simplr 731 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  A  e.  W )
7 eqid 2296 . . . . 5  |-  ( Ms  A )  =  ( Ms  A )
8 eqid 2296 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
91, 8mgpbas 15347 . . . . 5  |-  ( Base `  R )  =  (
Base `  M )
107, 9ressid2 13212 . . . 4  |-  ( ( ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  M )
112, 5, 6, 10syl3anc 1182 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  M )
12 simpll 730 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  R  e.  V )
13 mgpress.1 . . . . . 6  |-  S  =  ( Rs  A )
1413, 8ressid2 13212 . . . . 5  |-  ( ( ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  R )
152, 12, 6, 14syl3anc 1182 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  S  =  R )
1615fveq2d 5545 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  (mulGrp `  S
)  =  (mulGrp `  R ) )
171, 11, 163eqtr4a 2354 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  (mulGrp `  S ) )
18 eqid 2296 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18mgpval 15344 . . . 4  |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
2019oveq1i 5884 . . 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 447 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  -.  ( Base `  R )  C_  A )
224a1i 10 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  M  e.  _V )
23 simplr 731 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  A  e.  W )
247, 9ressval2 13213 . . . 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 1182 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
26 eqid 2296 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
27 eqid 2296 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2826, 27mgpval 15344 . . . . 5  |-  (mulGrp `  S )  =  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )
29 simpll 730 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  R  e.  V )
3013, 8ressval2 13213 . . . . . . 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 1182 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  S  =  ( R sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
3213, 18ressmulr 13277 . . . . . . . . 9  |-  ( A  e.  W  ->  ( .r `  R )  =  ( .r `  S
) )
3332eqcomd 2301 . . . . . . . 8  |-  ( A  e.  W  ->  ( .r `  S )  =  ( .r `  R
) )
3433ad2antlr 707 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( .r `  S )  =  ( .r `  R
) )
3534opeq2d 3819 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  <. ( +g  `  ndx ) ,  ( .r `  S
) >.  =  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
3631, 35oveq12d 5892 . . . . 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 2340 . . . 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 9947 . . . . . . 7  |-  1  =/=  2
3938necomi 2541 . . . . . 6  |-  2  =/=  1
40 plusgndx 13258 . . . . . . 7  |-  ( +g  ` 
ndx )  =  2
41 basendx 13209 . . . . . . 7  |-  ( Base `  ndx )  =  1
4240, 41neeq12i 2471 . . . . . 6  |-  ( ( +g  `  ndx )  =/=  ( Base `  ndx ) 
<->  2  =/=  1 )
4339, 42mpbir 200 . . . . 5  |-  ( +g  ` 
ndx )  =/=  ( Base `  ndx )
44 fvex 5555 . . . . . 6  |-  ( .r
`  R )  e. 
_V
45 fvex 5555 . . . . . . 7  |-  ( Base `  R )  e.  _V
4645inex2 4172 . . . . . 6  |-  ( A  i^i  ( Base `  R
) )  e.  _V
47 fvex 5555 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  _V
48 fvex 5555 . . . . . . 7  |-  ( Base `  ndx )  e.  _V
4947, 48setscom 13192 . . . . . 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 666 . . . . 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 643 . . . 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 2331 . . 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 2354 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  (mulGrp `  S ) )
5417, 53pm2.61dan 766 1  |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    i^i cin 3164    C_ wss 3165   <.cop 3656   ` cfv 5271  (class class class)co 5874   1c1 8754   2c2 9811   ndxcnx 13161   sSet csts 13162   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225  mulGrpcmgp 15341
This theorem is referenced by:  subrgcrng  15565  subrgsubm  15574  resrhm  15590  lgsqrlem1  20596  lgseisenlem4  20607  dchrisum0flblem1  20673  xrge0iifmhm  23336  xrge0pluscn  23337  xrge0tmd  23343
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  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-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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-mgp 15342
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