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Theorem dsmmsubg 27209
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 2284 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2284 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 5749 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 642 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2283 . . . . . 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 27201 . . . . 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 15 . . . 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 3258 . . . . 5  |-  { a  e.  ( Base `  ( S X_s R ) )  |  { b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  C_  ( Base `  ( S X_s R ) )
1211a1i 10 . . . 4  |-  ( ph  ->  { a  e.  (
Base `  ( S X_s R ) )  |  {
b  e.  dom  R  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin }  C_  ( Base `  ( S X_s R ) ) )
1310, 12eqsstr3d 3213 . . 3  |-  ( ph  ->  ( Base `  ( S  (+)m  R ) )  C_  ( Base `  ( S X_s R ) ) )
14 dsmmsubg.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
15 dsmmsubg.p . . . 4  |-  P  =  ( S X_s R )
1615fveq2i 5528 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1713, 14, 163sstr4g 3219 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
18 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
19 grpmnd 14494 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
2019ssriv 3184 . . . 4  |-  Grp  C_  Mnd
21 fss 5397 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
224, 20, 21sylancl 643 . . 3  |-  ( ph  ->  R : I --> Mnd )
23 eqid 2283 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2415, 14, 5, 18, 22, 23dsmm0cl 27206 . 2  |-  ( ph  ->  ( 0g `  P
)  e.  H )
2553ad2ant1 976 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  I  e.  W )
26183ad2ant1 976 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  S  e.  V )
27223ad2ant1 976 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  R :
I --> Mnd )
28 simp2 956 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  a  e.  H )
29 simp3 957 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  b  e.  H )
30 eqid 2283 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3115, 14, 25, 26, 27, 28, 29, 30dsmmacl 27207 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3215, 5, 18, 4prdsgrpd 14604 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3332adantr 451 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3417sselda 3180 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
35 eqid 2283 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
36 eqid 2283 . . . . 5  |-  ( inv g `  P )  =  ( inv g `  P )
3735, 36grpinvcl 14527 . . . 4  |-  ( ( P  e.  Grp  /\  a  e.  ( Base `  P ) )  -> 
( ( inv g `  P ) `  a
)  e.  ( Base `  P ) )
3833, 34, 37syl2anc 642 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( inv g `  P ) `  a
)  e.  ( Base `  P ) )
39 simpr 447 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  H )
40 eqid 2283 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
415adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
42 ffn 5389 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
434, 42syl 15 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
4443adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  R  Fn  I )
4515, 40, 35, 14, 41, 44dsmmelbas 27205 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  H  <->  ( a  e.  ( Base `  P
)  /\  { b  e.  I  |  (
a `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin ) ) )
4639, 45mpbid 201 . . . . 5  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  ( Base `  P )  /\  {
b  e.  I  |  ( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) }  e.  Fin ) )
4746simprd 449 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
485ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  I  e.  W )
4918ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  S  e.  V )
504ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  R : I --> Grp )
5134adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  a  e.  ( Base `  P
) )
52 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  b  e.  I )
5315, 48, 49, 50, 35, 36, 51, 52prdsinvgd2 27208 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( inv g `  P ) `  a
) `  b )  =  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) ) )
5453adantrr 697 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( inv g `  P ) `  a
) `  b )  =  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) ) )
55 fveq2 5525 . . . . . . . . 9  |-  ( ( a `  b )  =  ( 0g `  ( R `  b ) )  ->  ( ( inv g `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( inv g `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
5655ad2antll 709 . . . . . . . 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 ) ) ) )
57 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( R : I --> Grp  /\  b  e.  I )  ->  ( R `  b
)  e.  Grp )
584, 57sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
5958adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
60 eqid 2283 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
61 eqid 2283 . . . . . . . . . . 11  |-  ( inv g `  ( R `
 b ) )  =  ( inv g `  ( R `  b
) )
6260, 61grpinvid 14533 . . . . . . . . . 10  |-  ( ( R `  b )  e.  Grp  ->  (
( inv g `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6359, 62syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( inv g `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6463adantrr 697 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( ( inv g `  ( R `
 b ) ) `
 ( 0g `  ( R `  b ) ) )  =  ( 0g `  ( R `
 b ) ) )
6554, 56, 643eqtrd 2319 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( inv g `  P ) `  a
) `  b )  =  ( 0g `  ( R `  b ) ) )
6665expr 598 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( a `  b
)  =  ( 0g
`  ( R `  b ) )  -> 
( ( ( inv g `  P ) `
 a ) `  b )  =  ( 0g `  ( R `
 b ) ) ) )
6766necon3d 2484 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( inv g `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6867ss2rabdv 3254 . . . 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 ) ) } )
69 ssfi 7083 . . . 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 )
7047, 68, 69syl2anc 642 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( inv g `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin )
7115, 40, 35, 14, 41, 44dsmmelbas 27205 . . 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 ) ) )
7238, 70, 71mpbir2and 888 . 2  |-  ( (
ph  /\  a  e.  H )  ->  (
( inv g `  P ) `  a
)  e.  H )
731, 2, 3, 17, 24, 31, 72, 32issubgrpd2 15941 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    C_ wss 3152   dom cdm 4689    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   X_scprds 13346   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615    (+)m cdsmm 27197
This theorem is referenced by:  dsmmlss  27210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-dsmm 27198
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