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Theorem dsmmsubg 18643
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 2468 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2468 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2468 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 6144 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 661 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2467 . . . . . 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 18635 . . . . 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 3590 . . . 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 3560 . . 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 5875 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1612, 13, 153sstr4g 3550 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
17 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
18 grpmnd 15934 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
1918ssriv 3513 . . . 4  |-  Grp  C_  Mnd
20 fss 5745 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
214, 19, 20sylancl 662 . . 3  |-  ( ph  ->  R : I --> Mnd )
22 eqid 2467 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2314, 13, 5, 17, 21, 22dsmm0cl 18640 . 2  |-  ( ph  ->  ( 0g `  P
)  e.  H )
2453ad2ant1 1017 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  I  e.  W )
25173ad2ant1 1017 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  S  e.  V )
26213ad2ant1 1017 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  R :
I --> Mnd )
27 simp2 997 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  a  e.  H )
28 simp3 998 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  b  e.  H )
29 eqid 2467 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3014, 13, 24, 25, 26, 27, 28, 29dsmmacl 18641 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3114, 5, 17, 4prdsgrpd 16051 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3231adantr 465 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3316sselda 3509 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
34 eqid 2467 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
35 eqid 2467 . . . . 5  |-  ( invg `  P )  =  ( invg `  P )
3634, 35grpinvcl 15967 . . . 4  |-  ( ( P  e.  Grp  /\  a  e.  ( Base `  P ) )  -> 
( ( invg `  P ) `  a
)  e.  ( Base `  P ) )
3732, 33, 36syl2anc 661 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( invg `  P ) `  a
)  e.  ( Base `  P ) )
38 simpr 461 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  H )
39 eqid 2467 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
405adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
41 ffn 5737 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
424, 41syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
4342adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  R  Fn  I )
4414, 39, 34, 13, 40, 43dsmmelbas 18639 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  H  <->  ( a  e.  ( Base `  P
)  /\  { b  e.  I  |  (
a `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin ) ) )
4538, 44mpbid 210 . . . . 5  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  ( Base `  P )  /\  {
b  e.  I  |  ( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) }  e.  Fin ) )
4645simprd 463 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
475ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  I  e.  W )
4817ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  S  e.  V )
494ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  R : I --> Grp )
5033adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  a  e.  ( Base `  P
) )
51 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  b  e.  I )
5214, 47, 48, 49, 34, 35, 50, 51prdsinvgd2 18642 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( invg `  P ) `  a
) `  b )  =  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) ) )
5352adantrr 716 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( invg `  P ) `  a
) `  b )  =  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) ) )
54 fveq2 5872 . . . . . . . . 9  |-  ( ( a `  b )  =  ( 0g `  ( R `  b ) )  ->  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( invg `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
5554ad2antll 728 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( invg `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
564ffvelrnda 6032 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
5756adantlr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
58 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
59 eqid 2467 . . . . . . . . . . 11  |-  ( invg `  ( R `
 b ) )  =  ( invg `  ( R `  b
) )
6058, 59grpinvid 15973 . . . . . . . . . 10  |-  ( ( R `  b )  e.  Grp  ->  (
( invg `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6157, 60syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( invg `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6261adantrr 716 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( ( invg `  ( R `
 b ) ) `
 ( 0g `  ( R `  b ) ) )  =  ( 0g `  ( R `
 b ) ) )
6353, 55, 623eqtrd 2512 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( invg `  P ) `  a
) `  b )  =  ( 0g `  ( R `  b ) ) )
6463expr 615 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( a `  b
)  =  ( 0g
`  ( R `  b ) )  -> 
( ( ( invg `  P ) `
 a ) `  b )  =  ( 0g `  ( R `
 b ) ) ) )
6564necon3d 2691 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( invg `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6665ss2rabdv 3586 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( invg `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  C_  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) } )
67 ssfi 7752 . . . 4  |-  ( ( { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin  /\  {
b  e.  I  |  ( ( ( invg `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) ) } 
C_  { b  e.  I  |  ( a `
 b )  =/=  ( 0g `  ( R `  b )
) } )  ->  { b  e.  I  |  ( ( ( invg `  P
) `  a ) `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
6846, 66, 67syl2anc 661 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( invg `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin )
6914, 39, 34, 13, 40, 43dsmmelbas 18639 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( ( invg `  P ) `  a
)  e.  H  <->  ( (
( invg `  P ) `  a
)  e.  ( Base `  P )  /\  {
b  e.  I  |  ( ( ( invg `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin ) ) )
7037, 68, 69mpbir2and 920 . 2  |-  ( (
ph  /\  a  e.  H )  ->  (
( invg `  P ) `  a
)  e.  H )
711, 2, 3, 16, 23, 30, 70, 31issubgrpd2 16089 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118    C_ wss 3481   dom cdm 5005    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   0gc0g 14712   X_scprds 14718   Mndcmnd 15793   Grpcgrp 15925   invgcminusg 15926  SubGrpcsubg 16067    (+)m cdsmm 18631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-subg 16070  df-dsmm 18632
This theorem is referenced by:  dsmmlss  18644
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