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Theorem dsmmsubg 19304
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 2423 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2423 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2423 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 6153 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 665 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2422 . . . . . 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 19296 . . . . 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 17 . . . 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 3546 . . . 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 3515 . . 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 5884 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1612, 13, 153sstr4g 3505 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
17 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
18 grpmnd 16677 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
1918ssriv 3468 . . . 4  |-  Grp  C_  Mnd
20 fss 5754 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
214, 19, 20sylancl 666 . . 3  |-  ( ph  ->  R : I --> Mnd )
22 eqid 2422 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2314, 13, 5, 17, 21, 22dsmm0cl 19301 . 2  |-  ( ph  ->  ( 0g `  P
)  e.  H )
2453ad2ant1 1026 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  I  e.  W )
25173ad2ant1 1026 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  S  e.  V )
26213ad2ant1 1026 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  R :
I --> Mnd )
27 simp2 1006 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  a  e.  H )
28 simp3 1007 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  b  e.  H )
29 eqid 2422 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3014, 13, 24, 25, 26, 27, 28, 29dsmmacl 19302 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3114, 5, 17, 4prdsgrpd 16794 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3231adantr 466 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3316sselda 3464 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
34 eqid 2422 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
35 eqid 2422 . . . . 5  |-  ( invg `  P )  =  ( invg `  P )
3634, 35grpinvcl 16710 . . . 4  |-  ( ( P  e.  Grp  /\  a  e.  ( Base `  P ) )  -> 
( ( invg `  P ) `  a
)  e.  ( Base `  P ) )
3732, 33, 36syl2anc 665 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  (
( invg `  P ) `  a
)  e.  ( Base `  P ) )
38 simpr 462 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  H )
39 eqid 2422 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
405adantr 466 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
41 ffn 5746 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
424, 41syl 17 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
4342adantr 466 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  R  Fn  I )
4414, 39, 34, 13, 40, 43dsmmelbas 19300 . . . . . 6  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  H  <->  ( a  e.  ( Base `  P
)  /\  { b  e.  I  |  (
a `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin ) ) )
4538, 44mpbid 213 . . . . 5  |-  ( (
ph  /\  a  e.  H )  ->  (
a  e.  ( Base `  P )  /\  {
b  e.  I  |  ( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) }  e.  Fin ) )
4645simprd 464 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( a `  b )  =/=  ( 0g `  ( R `  b ) ) }  e.  Fin )
475ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  I  e.  W )
4817ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  S  e.  V )
494ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  R : I --> Grp )
5033adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  a  e.  ( Base `  P
) )
51 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  b  e.  I )
5214, 47, 48, 49, 34, 35, 50, 51prdsinvgd2 19303 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( invg `  P ) `  a
) `  b )  =  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) ) )
5352adantrr 721 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( invg `  P ) `  a
) `  b )  =  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) ) )
54 fveq2 5881 . . . . . . . . 9  |-  ( ( a `  b )  =  ( 0g `  ( R `  b ) )  ->  ( ( invg `  ( R `
 b ) ) `
 ( a `  b ) )  =  ( ( invg `  ( R `  b
) ) `  ( 0g `  ( R `  b ) ) ) )
5554ad2antll 733 . . . . . . . 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 6037 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
5756adantlr 719 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  ( R `  b )  e.  Grp )
58 eqid 2422 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
59 eqid 2422 . . . . . . . . . . 11  |-  ( invg `  ( R `
 b ) )  =  ( invg `  ( R `  b
) )
6058, 59grpinvid 16716 . . . . . . . . . 10  |-  ( ( R `  b )  e.  Grp  ->  (
( invg `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6157, 60syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( invg `  ( R `  b ) ) `  ( 0g
`  ( R `  b ) ) )  =  ( 0g `  ( R `  b ) ) )
6261adantrr 721 . . . . . . . 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 2467 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  H )  /\  (
b  e.  I  /\  ( a `  b
)  =  ( 0g
`  ( R `  b ) ) ) )  ->  ( (
( invg `  P ) `  a
) `  b )  =  ( 0g `  ( R `  b ) ) )
6463expr 618 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( a `  b
)  =  ( 0g
`  ( R `  b ) )  -> 
( ( ( invg `  P ) `
 a ) `  b )  =  ( 0g `  ( R `
 b ) ) ) )
6564necon3d 2644 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( invg `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6665ss2rabdv 3542 . . . 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 7801 . . . 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 665 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  { b  e.  I  |  ( ( ( invg `  P ) `  a
) `  b )  =/=  ( 0g `  ( R `  b )
) }  e.  Fin )
6914, 39, 34, 13, 40, 43dsmmelbas 19300 . . 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 930 . 2  |-  ( (
ph  /\  a  e.  H )  ->  (
( invg `  P ) `  a
)  e.  H )
711, 2, 3, 16, 23, 30, 70, 31issubgrpd2 16832 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   {crab 2775   _Vcvv 3080    C_ wss 3436   dom cdm 4853    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7580   Basecbs 15120   ↾s cress 15121   +g cplusg 15189   0gc0g 15337   X_scprds 15343   Mndcmnd 16534   Grpcgrp 16668   invgcminusg 16669  SubGrpcsubg 16810    (+)m cdsmm 19292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-hom 15213  df-cco 15214  df-0g 15339  df-prds 15345  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-grp 16672  df-minusg 16673  df-subg 16813  df-dsmm 19293
This theorem is referenced by:  dsmmlss  19305
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