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Theorem dsmmsubg 27177
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 2436 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2436 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2436 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 5961 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 643 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2435 . . . . . 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 27169 . . . . 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 3420 . . . 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 3391 . . 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 5723 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1612, 13, 153sstr4g 3381 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
17 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
18 grpmnd 14809 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
1918ssriv 3344 . . . 4  |-  Grp  C_  Mnd
20 fss 5591 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
214, 19, 20sylancl 644 . . 3  |-  ( ph  ->  R : I --> Mnd )
22 eqid 2435 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2314, 13, 5, 17, 21, 22dsmm0cl 27174 . 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 2435 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3014, 13, 24, 25, 26, 27, 28, 29dsmmacl 27175 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3114, 5, 17, 4prdsgrpd 14919 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3231adantr 452 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3316sselda 3340 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
34 eqid 2435 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
35 eqid 2435 . . . . 5  |-  ( inv g `  P )  =  ( inv g `  P )
3634, 35grpinvcl 14842 . . . 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 2435 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
405adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
41 ffn 5583 . . . . . . . . 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 27173 . . . . . 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 27176 . . . . . . . . 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 5720 . . . . . . . . 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 5862 . . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
59 eqid 2435 . . . . . . . . . . 11  |-  ( inv g `  ( R `
 b ) )  =  ( inv g `  ( R `  b
) )
6058, 59grpinvid 14848 . . . . . . . . . 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 2471 . . . . . . 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 2636 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( inv g `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6665ss2rabdv 3416 . . . 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 7321 . . . 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 27173 . . 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 16252 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    C_ wss 3312   dom cdm 4870    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   X_scprds 13661   0gc0g 13715   Mndcmnd 14676   Grpcgrp 14677   inv gcminusg 14678  SubGrpcsubg 14930    (+)m cdsmm 27165
This theorem is referenced by:  dsmmlss  27178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-dsmm 27166
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