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Theorem subgint 15819
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 4253 . . . 4  |-  ( S  =/=  (/)  ->  |^| S  C_  U. S )
21adantl 466 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  U. S )
3 ssel2 3454 . . . . . . 7  |-  ( ( S  C_  (SubGrp `  G
)  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
43adantlr 714 . . . . . 6  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
5 eqid 2452 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
65subgss 15796 . . . . . 6  |-  ( g  e.  (SubGrp `  G
)  ->  g  C_  ( Base `  G )
)
74, 6syl 16 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  g  C_  ( Base `  G
) )
87ralrimiva 2827 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  g  C_  ( Base `  G )
)
9 unissb 4226 . . . 4  |-  ( U. S  C_  ( Base `  G
)  <->  A. g  e.  S  g  C_  ( Base `  G
) )
108, 9sylibr 212 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  U. S  C_  ( Base `  G )
)
112, 10sstrd 3469 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  ( Base `  G )
)
12 eqid 2452 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1312subg0cl 15803 . . . . . 6  |-  ( g  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  g )
144, 13syl 16 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  ( 0g `  G )  e.  g )
1514ralrimiva 2827 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  ( 0g `  G )  e.  g )
16 fvex 5804 . . . . 5  |-  ( 0g
`  G )  e. 
_V
1716elint2 4238 . . . 4  |-  ( ( 0g `  G )  e.  |^| S  <->  A. g  e.  S  ( 0g `  G )  e.  g )
1815, 17sylibr 212 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  ( 0g `  G )  e.  |^| S )
19 ne0i 3746 . . 3  |-  ( ( 0g `  G )  e.  |^| S  ->  |^| S  =/=  (/) )
2018, 19syl 16 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  =/=  (/) )
214adantlr 714 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
22 simprl 755 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
23 elinti 4240 . . . . . . . . . . 11  |-  ( x  e.  |^| S  ->  (
g  e.  S  ->  x  e.  g )
)
2423imp 429 . . . . . . . . . 10  |-  ( ( x  e.  |^| S  /\  g  e.  S
)  ->  x  e.  g )
2522, 24sylan 471 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  x  e.  g )
26 simprr 756 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
27 elinti 4240 . . . . . . . . . . 11  |-  ( y  e.  |^| S  ->  (
g  e.  S  -> 
y  e.  g ) )
2827imp 429 . . . . . . . . . 10  |-  ( ( y  e.  |^| S  /\  g  e.  S
)  ->  y  e.  g )
2926, 28sylan 471 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  y  e.  g )
30 eqid 2452 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3130subgcl 15805 . . . . . . . . 9  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g  /\  y  e.  g )  ->  (
x ( +g  `  G
) y )  e.  g )
3221, 25, 29, 31syl3anc 1219 . . . . . . . 8  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  (
x ( +g  `  G
) y )  e.  g )
3332ralrimiva 2827 . . . . . . 7  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
34 ovex 6220 . . . . . . . 8  |-  ( x ( +g  `  G
) y )  e. 
_V
3534elint2 4238 . . . . . . 7  |-  ( ( x ( +g  `  G
) y )  e. 
|^| S  <->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
3633, 35sylibr 212 . . . . . 6  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( +g  `  G
) y )  e. 
|^| S )
3736anassrs 648 . . . . 5  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  y  e.  |^| S )  ->  ( x ( +g  `  G ) y )  e.  |^| S )
3837ralrimiva 2827 . . . 4  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. y  e.  |^| S
( x ( +g  `  G ) y )  e.  |^| S )
394adantlr 714 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G ) )
4024adantll 713 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  x  e.  g )
41 eqid 2452 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
4241subginvcl 15804 . . . . . . 7  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g )  ->  (
( invg `  G ) `  x
)  e.  g )
4339, 40, 42syl2anc 661 . . . . . 6  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  ( ( invg `  G ) `  x
)  e.  g )
4443ralrimiva 2827 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. g  e.  S  ( ( invg `  G ) `  x
)  e.  g )
45 fvex 5804 . . . . . 6  |-  ( ( invg `  G
) `  x )  e.  _V
4645elint2 4238 . . . . 5  |-  ( ( ( invg `  G ) `  x
)  e.  |^| S  <->  A. g  e.  S  ( ( invg `  G ) `  x
)  e.  g )
4744, 46sylibr 212 . . . 4  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  -> 
( ( invg `  G ) `  x
)  e.  |^| S
)
4838, 47jca 532 . . 3  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  -> 
( A. y  e. 
|^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( invg `  G ) `  x
)  e.  |^| S
) )
4948ralrimiva 2827 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S ( A. y  e.  |^| S ( x ( +g  `  G
) y )  e. 
|^| S  /\  (
( invg `  G ) `  x
)  e.  |^| S
) )
50 ssn0 3773 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  (SubGrp `  G
)  =/=  (/) )
51 n0 3749 . . . 4  |-  ( (SubGrp `  G )  =/=  (/)  <->  E. g 
g  e.  (SubGrp `  G ) )
52 subgrcl 15800 . . . . 5  |-  ( g  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5352exlimiv 1689 . . . 4  |-  ( E. g  g  e.  (SubGrp `  G )  ->  G  e.  Grp )
5451, 53sylbi 195 . . 3  |-  ( (SubGrp `  G )  =/=  (/)  ->  G  e.  Grp )
555, 30, 41issubg2 15810 . . 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  /\  (
( invg `  G ) `  x
)  e.  |^| S
) ) ) )
5650, 54, 553syl 20 . 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  /\  (
( invg `  G ) `  x
)  e.  |^| S
) ) ) )
5711, 20, 49, 56mpbir3and 1171 1  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   E.wex 1587    e. wcel 1758    =/= wne 2645   A.wral 2796    C_ wss 3431   (/)c0 3740   U.cuni 4194   |^|cint 4231   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   0gc0g 14492   Grpcgrp 15524   invgcminusg 15525  SubGrpcsubg 15789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-0g 14494  df-mnd 15529  df-grp 15659  df-minusg 15660  df-subg 15792
This theorem is referenced by:  subrgint  17005
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