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Theorem dsmmsubg 26871
Description: The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmsubg.p  |-  P  =  ( S X_s R )
dsmmsubg.h  |-  H  =  ( Base `  ( S  (+)m  R ) )
dsmmsubg.i  |-  ( ph  ->  I  e.  W )
dsmmsubg.s  |-  ( ph  ->  S  e.  V )
dsmmsubg.r  |-  ( ph  ->  R : I --> Grp )
Assertion
Ref Expression
dsmmsubg  |-  ( ph  ->  H  e.  (SubGrp `  P ) )

Proof of Theorem dsmmsubg
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2381 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2381 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2381 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 5901 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 643 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2380 . . . . . 6  |-  { a  e.  ( Base `  ( S X_s R ) )  |  { b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  =  { a  e.  (
Base `  ( S X_s R ) )  |  {
b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }
98dsmmbase 26863 . . . . 5  |-  ( R  e.  _V  ->  { a  e.  ( Base `  ( S X_s R ) )  |  { b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
107, 9syl 16 . . . 4  |-  ( ph  ->  { a  e.  (
Base `  ( S X_s R ) )  |  {
b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
11 ssrab2 3364 . . . 4  |-  { a  e.  ( Base `  ( S X_s R ) )  |  { b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  C_  ( Base `  ( S X_s R ) )
1210, 11syl6eqssr 3335 . . 3  |-  ( ph  ->  ( Base `  ( S  (+)m  R ) )  C_  ( Base `  ( S X_s R ) ) )
13 dsmmsubg.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
14 dsmmsubg.p . . . 4  |-  P  =  ( S X_s R )
1514fveq2i 5664 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1612, 13, 153sstr4g 3325 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
17 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
18 grpmnd 14737 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
1918ssriv 3288 . . . 4  |-  Grp  C_  Mnd
20 fss 5532 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
214, 19, 20sylancl 644 . . 3  |-  ( ph  ->  R : I --> Mnd )
22 eqid 2380 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2314, 13, 5, 17, 21, 22dsmm0cl 26868 . 2  |-  ( ph  ->  ( 0g `  P
)  e.  H )
2453ad2ant1 978 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  I  e.  W )
25173ad2ant1 978 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  S  e.  V )
26213ad2ant1 978 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  R :
I --> Mnd )
27 simp2 958 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  a  e.  H )
28 simp3 959 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  b  e.  H )
29 eqid 2380 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3014, 13, 24, 25, 26, 27, 28, 29dsmmacl 26869 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3114, 5, 17, 4prdsgrpd 14847 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3231adantr 452 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3316sselda 3284 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
34 eqid 2380 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
35 eqid 2380 . . . . 5  |-  ( inv g `  P )  =  ( inv g `  P )
3634, 35grpinvcl 14770 . . . 4  |-  ( ( P  e.  Grp  /\  a  e.  ( Base `  P ) )  -> 
( ( inv g `  P ) `  a
)  e.  ( Base `  P ) )
3732, 33, 36syl2anc 643 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( inv g `  P ) `  a
)  e.  ( Base `  P ) )
38 simpr 448 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  H )
39 eqid 2380 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
405adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
41 ffn 5524 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
424, 41syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
4342adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  R  Fn  I )
4414, 39, 34, 13, 40, 43dsmmelbas 26867 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  H  <->  ( a  e.  ( Base `  P
)  /\  { b  e.  I  |  (
a `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin ) ) )
4538, 44mpbid 202 . . . . 5  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  ( Base `  P )  /\  {
b  e.  I  |  ( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) }  e.  Fin ) )
4645simprd 450 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
475ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  I  e.  W )
4817ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  S  e.  V )
494ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  R : I --> Grp )
5033adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  a  e.  ( Base `  P
) )
51 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  b  e.  I )
5214, 47, 48, 49, 34, 35, 50, 51prdsinvgd2 26870 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( inv g `  P ) `  a
) `  b )  =  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) ) )
5352adantrr 698 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( inv g `  P ) `  a
) `  b )  =  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) ) )
54 fveq2 5661 . . . . . . . . 9  |-  ( ( a `  b )  =  ( 0g `  ( R `  b ) )  ->  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( inv g `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
5554ad2antll 710 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( inv g `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
564ffvelrnda 5802 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
5756adantlr 696 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
58 eqid 2380 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
59 eqid 2380 . . . . . . . . . . 11  |-  ( inv g `  ( R `
 b ) )  =  ( inv g `  ( R `  b
) )
6058, 59grpinvid 14776 . . . . . . . . . 10  |-  ( ( R `  b )  e.  Grp  ->  (
( inv g `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6157, 60syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( inv g `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6261adantrr 698 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( ( inv g `  ( R `
 b ) ) `
 ( 0g `  ( R `  b ) ) )  =  ( 0g `  ( R `
 b ) ) )
6353, 55, 623eqtrd 2416 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( inv g `  P ) `  a
) `  b )  =  ( 0g `  ( R `  b ) ) )
6463expr 599 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( a `  b
)  =  ( 0g
`  ( R `  b ) )  -> 
( ( ( inv g `  P ) `
 a ) `  b )  =  ( 0g `  ( R `
 b ) ) ) )
6564necon3d 2581 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( inv g `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6665ss2rabdv 3360 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( inv g `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  C_  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) } )
67 ssfi 7258 . . . 4  |-  ( ( { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin  /\  {
b  e.  I  |  ( ( ( inv g `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) ) } 
C_  { b  e.  I  |  ( a `
 b )  =/=  ( 0g `  ( R `  b )
) } )  ->  { b  e.  I  |  ( ( ( inv g `  P
) `  a ) `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
6846, 66, 67syl2anc 643 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( inv g `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin )
6914, 39, 34, 13, 40, 43dsmmelbas 26867 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( ( inv g `  P ) `  a
)  e.  H  <->  ( (
( inv g `  P ) `  a
)  e.  ( Base `  P )  /\  {
b  e.  I  |  ( ( ( inv g `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin ) ) )
7037, 68, 69mpbir2and 889 . 2  |-  ( (
ph  /\  a  e.  H )  ->  (
( inv g `  P ) `  a
)  e.  H )
711, 2, 3, 16, 23, 30, 70, 31issubgrpd2 16180 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   {crab 2646   _Vcvv 2892    C_ wss 3256   dom cdm 4811    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   Fincfn 7038   Basecbs 13389   ↾s cress 13390   +g cplusg 13449   X_scprds 13589   0gc0g 13643   Mndcmnd 14604   Grpcgrp 14605   inv gcminusg 14606  SubGrpcsubg 14858    (+)m cdsmm 26859
This theorem is referenced by:  dsmmlss  26872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-hom 13473  df-cco 13474  df-prds 13591  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-subg 14861  df-dsmm 26860
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