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Theorem mgpress 17812
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 468 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Base `  R )  C_  A
)
3 fvex 5889 . . . . . 6  |-  (mulGrp `  R )  e.  _V
41, 3eqeltri 2545 . . . . 5  |-  M  e. 
_V
54a1i 11 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  M  e.  _V )
6 simplr 770 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  A  e.  W )
7 eqid 2471 . . . . 5  |-  ( Ms  A )  =  ( Ms  A )
8 eqid 2471 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
91, 8mgpbas 17807 . . . . 5  |-  ( Base `  R )  =  (
Base `  M )
107, 9ressid2 15255 . . . 4  |-  ( ( ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  M )
112, 5, 6, 10syl3anc 1292 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  M )
12 simpll 768 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  R  e.  V )
13 mgpress.1 . . . . . 6  |-  S  =  ( Rs  A )
1413, 8ressid2 15255 . . . . 5  |-  ( ( ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  R )
152, 12, 6, 14syl3anc 1292 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  S  =  R )
1615fveq2d 5883 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  (mulGrp `  S
)  =  (mulGrp `  R ) )
171, 11, 163eqtr4a 2531 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  (mulGrp `  S ) )
18 eqid 2471 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18mgpval 17804 . . . 4  |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
2019oveq1i 6318 . . 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 468 . . . 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 770 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  A  e.  W )
247, 9ressval2 15256 . . . 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 1292 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
26 eqid 2471 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
27 eqid 2471 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2826, 27mgpval 17804 . . . . 5  |-  (mulGrp `  S )  =  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )
29 simpll 768 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  R  e.  V )
3013, 8ressval2 15256 . . . . . . 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 1292 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  S  =  ( R sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
3213, 18ressmulr 15328 . . . . . . . . 9  |-  ( A  e.  W  ->  ( .r `  R )  =  ( .r `  S
) )
3332eqcomd 2477 . . . . . . . 8  |-  ( A  e.  W  ->  ( .r `  S )  =  ( .r `  R
) )
3433ad2antlr 741 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( .r `  S )  =  ( .r `  R
) )
3534opeq2d 4165 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  <. ( +g  `  ndx ) ,  ( .r `  S
) >.  =  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
3631, 35oveq12d 6326 . . . . 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 2517 . . . 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 10845 . . . . . . 7  |-  1  =/=  2
3938necomi 2697 . . . . . 6  |-  2  =/=  1
40 plusgndx 15302 . . . . . . 7  |-  ( +g  ` 
ndx )  =  2
41 basendx 15251 . . . . . . 7  |-  ( Base `  ndx )  =  1
4240, 41neeq12i 2709 . . . . . 6  |-  ( ( +g  `  ndx )  =/=  ( Base `  ndx ) 
<->  2  =/=  1 )
4339, 42mpbir 214 . . . . 5  |-  ( +g  ` 
ndx )  =/=  ( Base `  ndx )
44 fvex 5889 . . . . . 6  |-  ( .r
`  R )  e. 
_V
45 fvex 5889 . . . . . . 7  |-  ( Base `  R )  e.  _V
4645inex2 4538 . . . . . 6  |-  ( A  i^i  ( Base `  R
) )  e.  _V
47 fvex 5889 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  _V
48 fvex 5889 . . . . . . 7  |-  ( Base `  ndx )  e.  _V
4947, 48setscom 15231 . . . . . 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 699 . . . . 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 675 . . . 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 2508 . . 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 2531 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  (mulGrp `  S ) )
5417, 53pm2.61dan 808 1  |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    i^i cin 3389    C_ wss 3390   <.cop 3965   ` cfv 5589  (class class class)co 6308   1c1 9558   2c2 10681   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   .rcmulr 15269  mulGrpcmgp 17801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-mgp 17802
This theorem is referenced by:  subrgcrng  18090  subrgsubm  18099  resrhm  18115  nn0srg  19114  rge0srg  19115  zringmpg  19140  m2cpmmhm  19846  rdivmuldivd  28628  xrge0iifmhm  28819  xrge0pluscn  28820  xrge0tmd  28826
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