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Theorem mgpress 17733
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 462 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Base `  R )  C_  A
)
3 fvex 5891 . . . . . 6  |-  (mulGrp `  R )  e.  _V
41, 3eqeltri 2503 . . . . 5  |-  M  e. 
_V
54a1i 11 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  M  e.  _V )
6 simplr 760 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  A  e.  W )
7 eqid 2422 . . . . 5  |-  ( Ms  A )  =  ( Ms  A )
8 eqid 2422 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
91, 8mgpbas 17728 . . . . 5  |-  ( Base `  R )  =  (
Base `  M )
107, 9ressid2 15176 . . . 4  |-  ( ( ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  M )
112, 5, 6, 10syl3anc 1264 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  M )
12 simpll 758 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  R  e.  V )
13 mgpress.1 . . . . . 6  |-  S  =  ( Rs  A )
1413, 8ressid2 15176 . . . . 5  |-  ( ( ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  R )
152, 12, 6, 14syl3anc 1264 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  S  =  R )
1615fveq2d 5885 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  (mulGrp `  S
)  =  (mulGrp `  R ) )
171, 11, 163eqtr4a 2489 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  (mulGrp `  S ) )
18 eqid 2422 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18mgpval 17725 . . . 4  |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
2019oveq1i 6315 . . 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 462 . . . 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 760 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  A  e.  W )
247, 9ressval2 15177 . . . 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 1264 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
26 eqid 2422 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
27 eqid 2422 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2826, 27mgpval 17725 . . . . 5  |-  (mulGrp `  S )  =  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )
29 simpll 758 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  R  e.  V )
3013, 8ressval2 15177 . . . . . . 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 1264 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  S  =  ( R sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
3213, 18ressmulr 15249 . . . . . . . . 9  |-  ( A  e.  W  ->  ( .r `  R )  =  ( .r `  S
) )
3332eqcomd 2430 . . . . . . . 8  |-  ( A  e.  W  ->  ( .r `  S )  =  ( .r `  R
) )
3433ad2antlr 731 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( .r `  S )  =  ( .r `  R
) )
3534opeq2d 4194 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  <. ( +g  `  ndx ) ,  ( .r `  S
) >.  =  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
3631, 35oveq12d 6323 . . . . 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 2475 . . . 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 10829 . . . . . . 7  |-  1  =/=  2
3938necomi 2690 . . . . . 6  |-  2  =/=  1
40 plusgndx 15223 . . . . . . 7  |-  ( +g  ` 
ndx )  =  2
41 basendx 15172 . . . . . . 7  |-  ( Base `  ndx )  =  1
4240, 41neeq12i 2709 . . . . . 6  |-  ( ( +g  `  ndx )  =/=  ( Base `  ndx ) 
<->  2  =/=  1 )
4339, 42mpbir 212 . . . . 5  |-  ( +g  ` 
ndx )  =/=  ( Base `  ndx )
44 fvex 5891 . . . . . 6  |-  ( .r
`  R )  e. 
_V
45 fvex 5891 . . . . . . 7  |-  ( Base `  R )  e.  _V
4645inex2 4566 . . . . . 6  |-  ( A  i^i  ( Base `  R
) )  e.  _V
47 fvex 5891 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  _V
48 fvex 5891 . . . . . . 7  |-  ( Base `  ndx )  e.  _V
4947, 48setscom 15152 . . . . . 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 689 . . . . 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 666 . . . 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 2466 . . 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 2489 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  (mulGrp `  S ) )
5417, 53pm2.61dan 798 1  |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080    i^i cin 3435    C_ wss 3436   <.cop 4004   ` cfv 5601  (class class class)co 6305   1c1 9547   2c2 10666   ndxcnx 15117   sSet csts 15118   Basecbs 15120   ↾s cress 15121   +g cplusg 15189   .rcmulr 15190  mulGrpcmgp 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-mgp 17723
This theorem is referenced by:  subrgcrng  18011  subrgsubm  18020  resrhm  18036  nn0srg  19035  rge0srg  19036  zringmpg  19061  m2cpmmhm  19767  rdivmuldivd  28562  xrge0iifmhm  28753  xrge0pluscn  28754  xrge0tmd  28760
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