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Theorem subgint 16094
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 4290 . . . 4  |-  ( S  =/=  (/)  ->  |^| S  C_  U. S )
21adantl 466 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  U. S )
3 ssel2 3481 . . . . . . 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 2441 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
65subgss 16071 . . . . . 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 2855 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  g  C_  ( Base `  G )
)
9 unissb 4262 . . . 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 3496 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  ( Base `  G )
)
12 eqid 2441 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1312subg0cl 16078 . . . . . 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 2855 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  ( 0g `  G )  e.  g )
16 fvex 5862 . . . . 5  |-  ( 0g
`  G )  e. 
_V
1716elint2 4274 . . . 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 3773 . . 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 4276 . . . . . . . . . . 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 4276 . . . . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3130subgcl 16080 . . . . . . . . 9  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g  /\  y  e.  g )  ->  (
x ( +g  `  G
) y )  e.  g )
3221, 25, 29, 31syl3anc 1227 . . . . . . . 8  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  (
x ( +g  `  G
) y )  e.  g )
3332ralrimiva 2855 . . . . . . 7  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
34 ovex 6305 . . . . . . . 8  |-  ( x ( +g  `  G
) y )  e. 
_V
3534elint2 4274 . . . . . . 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 2855 . . . 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 2441 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
4241subginvcl 16079 . . . . . . 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 2855 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. g  e.  S  ( ( invg `  G ) `  x
)  e.  g )
45 fvex 5862 . . . . . 6  |-  ( ( invg `  G
) `  x )  e.  _V
4645elint2 4274 . . . . 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 2855 . 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 3800 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  (SubGrp `  G
)  =/=  (/) )
51 n0 3776 . . . 4  |-  ( (SubGrp `  G )  =/=  (/)  <->  E. g 
g  e.  (SubGrp `  G ) )
52 subgrcl 16075 . . . . 5  |-  ( g  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5352exlimiv 1707 . . . 4  |-  ( E. g  g  e.  (SubGrp `  G )  ->  G  e.  Grp )
5451, 53sylbi 195 . . 3  |-  ( (SubGrp `  G )  =/=  (/)  ->  G  e.  Grp )
555, 30, 41issubg2 16085 . . 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 1178 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 972   E.wex 1597    e. wcel 1802    =/= wne 2636   A.wral 2791    C_ wss 3458   (/)c0 3767   U.cuni 4230   |^|cint 4267   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   0gc0g 14709   Grpcgrp 15922   invgcminusg 15923  SubGrpcsubg 16064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-subg 16067
This theorem is referenced by:  subrgint  17319
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