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Theorem subgint 16342
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 4222 . . . 4  |-  ( S  =/=  (/)  ->  |^| S  C_  U. S )
21adantl 464 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  U. S )
3 ssel2 3412 . . . . . . 7  |-  ( ( S  C_  (SubGrp `  G
)  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
43adantlr 712 . . . . . 6  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
5 eqid 2382 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
65subgss 16319 . . . . . 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 2796 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  g  C_  ( Base `  G )
)
9 unissb 4194 . . . 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 3427 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  C_  ( Base `  G )
)
12 eqid 2382 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1312subg0cl 16326 . . . . . 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 2796 . . . 4  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  A. g  e.  S  ( 0g `  G )  e.  g )
16 fvex 5784 . . . . 5  |-  ( 0g
`  G )  e. 
_V
1716elint2 4206 . . . 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 3717 . . 3  |-  ( ( 0g `  G )  e.  |^| S  ->  |^| S  =/=  (/) )
2018, 19syl 16 . 2  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  =/=  (/) )
214adantlr 712 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G )
)
22 simprl 754 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
23 elinti 4208 . . . . . . . . . . 11  |-  ( x  e.  |^| S  ->  (
g  e.  S  ->  x  e.  g )
)
2423imp 427 . . . . . . . . . 10  |-  ( ( x  e.  |^| S  /\  g  e.  S
)  ->  x  e.  g )
2522, 24sylan 469 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  x  e.  g )
26 simprr 755 . . . . . . . . . 10  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
27 elinti 4208 . . . . . . . . . . 11  |-  ( y  e.  |^| S  ->  (
g  e.  S  -> 
y  e.  g ) )
2827imp 427 . . . . . . . . . 10  |-  ( ( y  e.  |^| S  /\  g  e.  S
)  ->  y  e.  g )
2926, 28sylan 469 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  y  e.  g )
30 eqid 2382 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3130subgcl 16328 . . . . . . . . 9  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g  /\  y  e.  g )  ->  (
x ( +g  `  G
) y )  e.  g )
3221, 25, 29, 31syl3anc 1226 . . . . . . . 8  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  g  e.  S )  ->  (
x ( +g  `  G
) y )  e.  g )
3332ralrimiva 2796 . . . . . . 7  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. g  e.  S  ( x
( +g  `  G ) y )  e.  g )
34 ovex 6224 . . . . . . . 8  |-  ( x ( +g  `  G
) y )  e. 
_V
3534elint2 4206 . . . . . . 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 646 . . . . 5  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  y  e.  |^| S )  ->  ( x ( +g  `  G ) y )  e.  |^| S )
3837ralrimiva 2796 . . . 4  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. y  e.  |^| S
( x ( +g  `  G ) y )  e.  |^| S )
394adantlr 712 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  g  e.  (SubGrp `  G ) )
4024adantll 711 . . . . . . 7  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  x  e.  g )
41 eqid 2382 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
4241subginvcl 16327 . . . . . . 7  |-  ( ( g  e.  (SubGrp `  G )  /\  x  e.  g )  ->  (
( invg `  G ) `  x
)  e.  g )
4339, 40, 42syl2anc 659 . . . . . 6  |-  ( ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  /\  g  e.  S )  ->  ( ( invg `  G ) `  x
)  e.  g )
4443ralrimiva 2796 . . . . 5  |-  ( ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  /\  x  e.  |^| S )  ->  A. g  e.  S  ( ( invg `  G ) `  x
)  e.  g )
45 fvex 5784 . . . . . 6  |-  ( ( invg `  G
) `  x )  e.  _V
4645elint2 4206 . . . . 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 530 . . 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 2796 . 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 3745 . . 3  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  (SubGrp `  G
)  =/=  (/) )
51 n0 3721 . . . 4  |-  ( (SubGrp `  G )  =/=  (/)  <->  E. g 
g  e.  (SubGrp `  G ) )
52 subgrcl 16323 . . . . 5  |-  ( g  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5352exlimiv 1730 . . . 4  |-  ( E. g  g  e.  (SubGrp `  G )  ->  G  e.  Grp )
5451, 53sylbi 195 . . 3  |-  ( (SubGrp `  G )  =/=  (/)  ->  G  e.  Grp )
555, 30, 41issubg2 16333 . . 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 1177 1  |-  ( ( S  C_  (SubGrp `  G
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   E.wex 1620    e. wcel 1826    =/= wne 2577   A.wral 2732    C_ wss 3389   (/)c0 3711   U.cuni 4163   |^|cint 4199   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171  SubGrpcsubg 16312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315
This theorem is referenced by:  subrgint  17564
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