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Theorem dsmmsubg 18137
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 2438 . 2  |-  ( ph  ->  ( Ps  H )  =  ( Ps  H ) )
2 eqidd 2438 . 2  |-  ( ph  ->  ( 0g `  P
)  =  ( 0g
`  P ) )
3 eqidd 2438 . 2  |-  ( ph  ->  ( +g  `  P
)  =  ( +g  `  P ) )
4 dsmmsubg.r . . . . . 6  |-  ( ph  ->  R : I --> Grp )
5 dsmmsubg.i . . . . . 6  |-  ( ph  ->  I  e.  W )
6 fex 5943 . . . . . 6  |-  ( ( R : I --> Grp  /\  I  e.  W )  ->  R  e.  _V )
74, 5, 6syl2anc 661 . . . . 5  |-  ( ph  ->  R  e.  _V )
8 eqid 2437 . . . . . 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 18129 . . . . 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 3430 . . . 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 3400 . . 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 5687 . . 3  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
1612, 13, 153sstr4g 3390 . 2  |-  ( ph  ->  H  C_  ( Base `  P ) )
17 dsmmsubg.s . . 3  |-  ( ph  ->  S  e.  V )
18 grpmnd 15539 . . . . 5  |-  ( a  e.  Grp  ->  a  e.  Mnd )
1918ssriv 3353 . . . 4  |-  Grp  C_  Mnd
20 fss 5560 . . . 4  |-  ( ( R : I --> Grp  /\  Grp  C_  Mnd )  ->  R : I --> Mnd )
214, 19, 20sylancl 662 . . 3  |-  ( ph  ->  R : I --> Mnd )
22 eqid 2437 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
2314, 13, 5, 17, 21, 22dsmm0cl 18134 . 2  |-  ( ph  ->  ( 0g `  P
)  e.  H )
2453ad2ant1 1009 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  I  e.  W )
25173ad2ant1 1009 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  S  e.  V )
26213ad2ant1 1009 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  R :
I --> Mnd )
27 simp2 989 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  a  e.  H )
28 simp3 990 . . 3  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  b  e.  H )
29 eqid 2437 . . 3  |-  ( +g  `  P )  =  ( +g  `  P )
3014, 13, 24, 25, 26, 27, 28, 29dsmmacl 18135 . 2  |-  ( (
ph  /\  a  e.  H  /\  b  e.  H
)  ->  ( a
( +g  `  P ) b )  e.  H
)
3114, 5, 17, 4prdsgrpd 15653 . . . . 5  |-  ( ph  ->  P  e.  Grp )
3231adantr 465 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  P  e.  Grp )
3316sselda 3349 . . . 4  |-  ( (
ph  /\  a  e.  H )  ->  a  e.  ( Base `  P
) )
34 eqid 2437 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
35 eqid 2437 . . . . 5  |-  ( invg `  P )  =  ( invg `  P )
3634, 35grpinvcl 15572 . . . 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 2437 . . . . . . 7  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
405adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  H )  ->  I  e.  W )
41 ffn 5552 . . . . . . . . 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 18133 . . . . . 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 18136 . . . . . . . . 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 5684 . . . . . . . . 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 5836 . . . . . . . . . . 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 2437 . . . . . . . . . . 11  |-  ( 0g
`  ( R `  b ) )  =  ( 0g `  ( R `  b )
)
59 eqid 2437 . . . . . . . . . . 11  |-  ( invg `  ( R `
 b ) )  =  ( invg `  ( R `  b
) )
6058, 59grpinvid 15578 . . . . . . . . . 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 2473 . . . . . . 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 2640 . . . . 5  |-  ( ( ( ph  /\  a  e.  H )  /\  b  e.  I )  ->  (
( ( ( invg `  P ) `
 a ) `  b )  =/=  ( 0g `  ( R `  b ) )  -> 
( a `  b
)  =/=  ( 0g
`  ( R `  b ) ) ) )
6665ss2rabdv 3426 . . . 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 7525 . . . 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 18133 . . 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 913 . 2  |-  ( (
ph  /\  a  e.  H )  ->  (
( invg `  P ) `  a
)  e.  H )
711, 2, 3, 16, 23, 30, 70, 31issubgrpd2 15686 1  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2600   {crab 2713   _Vcvv 2966    C_ wss 3321   dom cdm 4832    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   Fincfn 7302   Basecbs 14166   ↾s cress 14167   +g cplusg 14230   0gc0g 14370   X_scprds 14376   Mndcmnd 15401   Grpcgrp 15402   invgcminusg 15403  SubGrpcsubg 15664    (+)m cdsmm 18125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-hom 14254  df-cco 14255  df-0g 14372  df-prds 14378  df-mnd 15407  df-grp 15534  df-minusg 15535  df-subg 15667  df-dsmm 18126
This theorem is referenced by:  dsmmlss  18138
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