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Theorem subgint 14657
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 3900 . . . 4  |-  ( S  =/=  (/)  ->  |^| S  C_  U. S )
21adantl 452 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  U. S )
3 ssel2 3188 . . . . . . 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 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
65subgss 14638 . . . . . 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 2639 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  g  C_  ( Base `  G )
)
9 unissb 3873 . . . 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 3202 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  ( Base `  G )
)
12 eqid 2296 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1312subg0cl 14645 . . . . . 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 2639 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  ( 0g `  G )  e.  g )
16 fvex 5555 . . . . 5  |-  ( 0g
`  G )  e. 
_V
1716elint2 3885 . . . 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 3474 . . 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 3887 . . . . . . . . . . 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 3887 . . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3130subgcl 14647 . . . . . . . . 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 2639 . . . . . . 7  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
34 ovex 5899 . . . . . . . 8  |-  ( x ( +g  `  G
) y )  e. 
_V
3534elint2 3885 . . . . . . 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 2639 . . . 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 2296 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
4241subginvcl 14646 . . . . . . 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 2639 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. g  e.  S  ( ( inv g `  G ) `  x
)  e.  g )
45 fvex 5555 . . . . . 6  |-  ( ( inv g `  G
) `  x )  e.  _V
4645elint2 3885 . . . . 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 2639 . 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 3500 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  (SubGrp `  G
)  =/=  (/) )
51 n0 3477 . . . 4  |-  ( (SubGrp `  G )  =/=  (/)  <->  E. g 
g  e.  (SubGrp `  G ) )
52 subgrcl 14642 . . . . 5  |-  ( g  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5352exlimiv 1624 . . . 4  |-  ( E. g  g  e.  (SubGrp `  G )  ->  G  e.  Grp )
5451, 53sylbi 187 . . 3  |-  ( (SubGrp `  G )  =/=  (/)  ->  G  e.  Grp )
555, 30, 41issubg2 14652 . . 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 1531    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379  SubGrpcsubg 14631
This theorem is referenced by:  subrgint  15583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634
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