MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgint Unicode version

Theorem subgint 14641
Description: The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
Assertion
Ref Expression
subgint  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  G )
)

Proof of Theorem subgint
Dummy variables  x  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 intssuni 3884 . . . 4  |-  ( S  =/=  (/)  ->  |^| S  C_  U. S )
21adantl 452 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  U. S )
3 ssel2 3175 . . . . . . 7  |-  ( ( S  C_  (SubGrp `  G
)  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
43adantlr 695 . . . . . 6  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
5 eqid 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
65subgss 14622 . . . . . 6  |-  ( g  e.  (SubGrp `  G
)  ->  g  C_  ( Base `  G )
)
74, 6syl 15 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  g  C_  ( Base `  G
) )
87ralrimiva 2626 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  g  C_  ( Base `  G )
)
9 unissb 3857 . . . 4  |-  ( U. S  C_  ( Base `  G
)  <->  A. g  e.  S  g  C_  ( Base `  G
) )
108, 9sylibr 203 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  U. S  C_  ( Base `  G )
)
112, 10sstrd 3189 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  ( Base `  G )
)
12 eqid 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1312subg0cl 14629 . . . . . 6  |-  ( g  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  g )
144, 13syl 15 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  ( 0g `  G )  e.  g )
1514ralrimiva 2626 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  ( 0g `  G )  e.  g )
16 fvex 5539 . . . . 5  |-  ( 0g
`  G )  e. 
_V
1716elint2 3869 . . . 4  |-  ( ( 0g `  G )  e.  |^| S  <->  A. g  e.  S  ( 0g `  G )  e.  g )
1815, 17sylibr 203 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  ( 0g `  G )  e.  |^| S )
19 ne0i 3461 . . 3  |-  ( ( 0g `  G )  e.  |^| S  ->  |^| S  =/=  (/) )
2018, 19syl 15 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  =/=  (/) )
214adantlr 695 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
22 simprl 732 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
23 elinti 3871 . . . . . . . . . . 11  |-  ( x  e.  |^| S  ->  (
g  e.  S  ->  x  e.  g )
)
2423imp 418 . . . . . . . . . 10  |-  ( ( x  e.  |^| S  /\  g  e.  S
)  ->  x  e.  g )
2522, 24sylan 457 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  x  e.  g )
26 simprr 733 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
27 elinti 3871 . . . . . . . . . . 11  |-  ( y  e.  |^| S  ->  (
g  e.  S  -> 
y  e.  g ) )
2827imp 418 . . . . . . . . . 10  |-  ( ( y  e.  |^| S  /\  g  e.  S
)  ->  y  e.  g )
2926, 28sylan 457 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  y  e.  g )
30 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3130subgcl 14631 . . . . . . . . 9  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g  /\  y  e.  g )  ->  (
x ( +g  `  G
) y )  e.  g )
3221, 25, 29, 31syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  (
x ( +g  `  G
) y )  e.  g )
3332ralrimiva 2626 . . . . . . 7  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
34 ovex 5883 . . . . . . . 8  |-  ( x ( +g  `  G
) y )  e. 
_V
3534elint2 3869 . . . . . . 7  |-  ( ( x ( +g  `  G
) y )  e. 
|^| S  <->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
3633, 35sylibr 203 . . . . . 6  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( +g  `  G
) y )  e. 
|^| S )
3736anassrs 629 . . . . 5  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  y  e.  |^| S )  ->  ( x ( +g  `  G ) y )  e.  |^| S )
3837ralrimiva 2626 . . . 4  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. y  e.  |^| S
( x ( +g  `  G ) y )  e.  |^| S )
394adantlr 695 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G ) )
4024adantll 694 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  x  e.  g )
41 eqid 2283 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
4241subginvcl 14630 . . . . . . 7  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g )  ->  (
( inv g `  G ) `  x
)  e.  g )
4339, 40, 42syl2anc 642 . . . . . 6  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  ( ( inv g `  G ) `  x
)  e.  g )
4443ralrimiva 2626 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. g  e.  S  ( ( inv g `  G ) `  x
)  e.  g )
45 fvex 5539 . . . . . 6  |-  ( ( inv g `  G
) `  x )  e.  _V
4645elint2 3869 . . . . 5  |-  ( ( ( inv g `  G ) `  x
)  e.  |^| S  <->  A. g  e.  S  ( ( inv g `  G ) `  x
)  e.  g )
4744, 46sylibr 203 . . . 4  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  -> 
( ( inv g `  G ) `  x
)  e.  |^| S
)
4838, 47jca 518 . . 3  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  -> 
( A. y  e. 
|^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( inv g `  G ) `  x
)  e.  |^| S
) )
4948ralrimiva 2626 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S ( A. y  e.  |^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( inv g `  G ) `  x
)  e.  |^| S
) )
50 ssn0 3487 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  (SubGrp `  G
)  =/=  (/) )
51 n0 3464 . . . 4  |-  ( (SubGrp `  G )  =/=  (/)  <->  E. g 
g  e.  (SubGrp `  G ) )
52 subgrcl 14626 . . . . 5  |-  ( g  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5352exlimiv 1666 . . . 4  |-  ( E. g  g  e.  (SubGrp `  G )  ->  G  e.  Grp )
5451, 53sylbi 187 . . 3  |-  ( (SubGrp `  G )  =/=  (/)  ->  G  e.  Grp )
555, 30, 41issubg2 14636 . . 3  |-  ( G  e.  Grp  ->  ( |^| S  e.  (SubGrp `  G )  <->  ( |^| S  C_  ( Base `  G
)  /\  |^| S  =/=  (/)  /\  A. x  e. 
|^| S ( A. y  e.  |^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( inv g `  G ) `  x
)  e.  |^| S
) ) ) )
5650, 54, 553syl 18 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  ( |^| S  e.  (SubGrp `  G
)  <->  ( |^| S  C_  ( Base `  G
)  /\  |^| S  =/=  (/)  /\  A. x  e. 
|^| S ( A. y  e.  |^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( inv g `  G ) `  x
)  e.  |^| S
) ) ) )
5711, 20, 49, 56mpbir3and 1135 1  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615
This theorem is referenced by:  subrgint  15567
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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618
  Copyright terms: Public domain W3C validator