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Theorem mgpress 17474
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 5861 . . . . . 6  |-  (mulGrp `  R )  e.  _V
41, 3eqeltri 2488 . . . . 5  |-  M  e. 
_V
54a1i 11 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  M  e.  _V )
6 simplr 756 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  A  e.  W )
7 eqid 2404 . . . . 5  |-  ( Ms  A )  =  ( Ms  A )
8 eqid 2404 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
91, 8mgpbas 17469 . . . . 5  |-  ( Base `  R )  =  (
Base `  M )
107, 9ressid2 14898 . . . 4  |-  ( ( ( Base `  R
)  C_  A  /\  M  e.  _V  /\  A  e.  W )  ->  ( Ms  A )  =  M )
112, 5, 6, 10syl3anc 1232 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  M )
12 simpll 754 . . . . 5  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  R  e.  V )
13 mgpress.1 . . . . . 6  |-  S  =  ( Rs  A )
1413, 8ressid2 14898 . . . . 5  |-  ( ( ( Base `  R
)  C_  A  /\  R  e.  V  /\  A  e.  W )  ->  S  =  R )
152, 12, 6, 14syl3anc 1232 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  S  =  R )
1615fveq2d 5855 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  (mulGrp `  S
)  =  (mulGrp `  R ) )
171, 11, 163eqtr4a 2471 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  ( Base `  R )  C_  A
)  ->  ( Ms  A
)  =  (mulGrp `  S ) )
18 eqid 2404 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18mgpval 17466 . . . 4  |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
2019oveq1i 6290 . . 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 756 . . . 4  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  A  e.  W )
247, 9ressval2 14899 . . . 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 1232 . . 3  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  ( M sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  R
) ) >. )
)
26 eqid 2404 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
27 eqid 2404 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2826, 27mgpval 17466 . . . . 5  |-  (mulGrp `  S )  =  ( S sSet  <. ( +g  `  ndx ) ,  ( .r `  S ) >. )
29 simpll 754 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  R  e.  V )
3013, 8ressval2 14899 . . . . . . 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 1232 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  S  =  ( R sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  R ) )
>. ) )
3213, 18ressmulr 14968 . . . . . . . . 9  |-  ( A  e.  W  ->  ( .r `  R )  =  ( .r `  S
) )
3332eqcomd 2412 . . . . . . . 8  |-  ( A  e.  W  ->  ( .r `  S )  =  ( .r `  R
) )
3433ad2antlr 727 . . . . . . 7  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( .r `  S )  =  ( .r `  R
) )
3534opeq2d 4168 . . . . . 6  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  <. ( +g  `  ndx ) ,  ( .r `  S
) >.  =  <. ( +g  `  ndx ) ,  ( .r `  R
) >. )
3631, 35oveq12d 6298 . . . . 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 2457 . . . 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 10791 . . . . . . 7  |-  1  =/=  2
3938necomi 2675 . . . . . 6  |-  2  =/=  1
40 plusgndx 14945 . . . . . . 7  |-  ( +g  ` 
ndx )  =  2
41 basendx 14895 . . . . . . 7  |-  ( Base `  ndx )  =  1
4240, 41neeq12i 2694 . . . . . 6  |-  ( ( +g  `  ndx )  =/=  ( Base `  ndx ) 
<->  2  =/=  1 )
4339, 42mpbir 211 . . . . 5  |-  ( +g  ` 
ndx )  =/=  ( Base `  ndx )
44 fvex 5861 . . . . . 6  |-  ( .r
`  R )  e. 
_V
45 fvex 5861 . . . . . . 7  |-  ( Base `  R )  e.  _V
4645inex2 4538 . . . . . 6  |-  ( A  i^i  ( Base `  R
) )  e.  _V
47 fvex 5861 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  _V
48 fvex 5861 . . . . . . 7  |-  ( Base `  ndx )  e.  _V
4947, 48setscom 14875 . . . . . 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 2448 . . 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 2471 . 2  |-  ( ( ( R  e.  V  /\  A  e.  W
)  /\  -.  ( Base `  R )  C_  A )  ->  ( Ms  A )  =  (mulGrp `  S ) )
5417, 53pm2.61dan 794 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 1407    e. wcel 1844    =/= wne 2600   _Vcvv 3061    i^i cin 3415    C_ wss 3416   <.cop 3980   ` cfv 5571  (class class class)co 6280   1c1 9525   2c2 10628   ndxcnx 14840   sSet csts 14841   Basecbs 14843   ↾s cress 14844   +g cplusg 14911   .rcmulr 14912  mulGrpcmgp 17463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-mgp 17464
This theorem is referenced by:  subrgcrng  17755  subrgsubm  17764  resrhm  17780  nn0srg  18808  rge0srg  18809  zringmpg  18831  m2cpmmhm  19540  rdivmuldivd  28247  xrge0iifmhm  28387  xrge0pluscn  28388  xrge0tmd  28394
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