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Theorem shintcli 21908
Description: Closure of intersection of a non-empty subset of  SH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
shintcl.1  |-  ( A 
C_  SH  /\  A  =/=  (/) )
Assertion
Ref Expression
shintcli  |-  |^| A  e.  SH

Proof of Theorem shintcli
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 shintcl.1 . . . . 5  |-  ( A 
C_  SH  /\  A  =/=  (/) )
21simpri 448 . . . 4  |-  A  =/=  (/)
3 n0 3464 . . . . 5  |-  ( A  =/=  (/)  <->  E. z  z  e.  A )
4 intss1 3877 . . . . . . 7  |-  ( z  e.  A  ->  |^| A  C_  z )
51simpli 444 . . . . . . . . 9  |-  A  C_  SH
65sseli 3176 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  SH )
7 shss 21789 . . . . . . . 8  |-  ( z  e.  SH  ->  z  C_ 
~H )
86, 7syl 15 . . . . . . 7  |-  ( z  e.  A  ->  z  C_ 
~H )
94, 8sstrd 3189 . . . . . 6  |-  ( z  e.  A  ->  |^| A  C_ 
~H )
109exlimiv 1666 . . . . 5  |-  ( E. z  z  e.  A  ->  |^| A  C_  ~H )
113, 10sylbi 187 . . . 4  |-  ( A  =/=  (/)  ->  |^| A  C_  ~H )
122, 11ax-mp 8 . . 3  |-  |^| A  C_ 
~H
13 ax-hv0cl 21583 . . . . . 6  |-  0h  e.  ~H
1413elexi 2797 . . . . 5  |-  0h  e.  _V
1514elint2 3869 . . . 4  |-  ( 0h  e.  |^| A  <->  A. z  e.  A  0h  e.  z )
16 sh0 21795 . . . . 5  |-  ( z  e.  SH  ->  0h  e.  z )
176, 16syl 15 . . . 4  |-  ( z  e.  A  ->  0h  e.  z )
1815, 17mprgbir 2613 . . 3  |-  0h  e.  |^| A
1912, 18pm3.2i 441 . 2  |-  ( |^| A  C_  ~H  /\  0h  e.  |^| A )
20 elinti 3871 . . . . . . . . 9  |-  ( x  e.  |^| A  ->  (
z  e.  A  ->  x  e.  z )
)
2120com12 27 . . . . . . . 8  |-  ( z  e.  A  ->  (
x  e.  |^| A  ->  x  e.  z ) )
22 elinti 3871 . . . . . . . . 9  |-  ( y  e.  |^| A  ->  (
z  e.  A  -> 
y  e.  z ) )
2322com12 27 . . . . . . . 8  |-  ( z  e.  A  ->  (
y  e.  |^| A  ->  y  e.  z ) )
24 shaddcl 21796 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
256, 24syl3an1 1215 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
26253expib 1154 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  +h  y )  e.  z ) )
2721, 23, 26syl2and 469 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  |^| A  /\  y  e.  |^| A )  ->  (
x  +h  y )  e.  z ) )
2827com12 27 . . . . . 6  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( z  e.  A  ->  ( x  +h  y )  e.  z ) )
2928ralrimiv 2625 . . . . 5  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  A. z  e.  A  ( x  +h  y )  e.  z )
30 ovex 5883 . . . . . 6  |-  ( x  +h  y )  e. 
_V
3130elint2 3869 . . . . 5  |-  ( ( x  +h  y )  e.  |^| A  <->  A. z  e.  A  ( x  +h  y )  e.  z )
3229, 31sylibr 203 . . . 4  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( x  +h  y )  e.  |^| A )
3332rgen2a 2609 . . 3  |-  A. x  e.  |^| A A. y  e.  |^| A ( x  +h  y )  e. 
|^| A
34 shmulcl 21797 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
356, 34syl3an1 1215 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
36353expib 1154 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  z )  ->  ( x  .h  y )  e.  z ) )
3723, 36sylan2d 468 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  |^| A
)  ->  ( x  .h  y )  e.  z ) )
3837com12 27 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( z  e.  A  ->  ( x  .h  y )  e.  z ) )
3938ralrimiv 2625 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  A. z  e.  A  ( x  .h  y
)  e.  z )
40 ovex 5883 . . . . . 6  |-  ( x  .h  y )  e. 
_V
4140elint2 3869 . . . . 5  |-  ( ( x  .h  y )  e.  |^| A  <->  A. z  e.  A  ( x  .h  y )  e.  z )
4239, 41sylibr 203 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( x  .h  y )  e.  |^| A )
4342rgen2 2639 . . 3  |-  A. x  e.  CC  A. y  e. 
|^| A ( x  .h  y )  e. 
|^| A
4433, 43pm3.2i 441 . 2  |-  ( A. x  e.  |^| A A. y  e.  |^| A ( x  +h  y )  e.  |^| A  /\  A. x  e.  CC  A. y  e.  |^| A ( x  .h  y )  e. 
|^| A )
45 issh2 21788 . 2  |-  ( |^| A  e.  SH  <->  ( ( |^| A  C_  ~H  /\  0h  e.  |^| A )  /\  ( A. x  e.  |^| A A. y  e.  |^| A ( x  +h  y )  e.  |^| A  /\  A. x  e.  CC  A. y  e. 
|^| A ( x  .h  y )  e. 
|^| A ) ) )
4619, 44, 45mpbir2an 886 1  |-  |^| A  e.  SH
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|cint 3862  (class class class)co 5858   CCcc 8735   ~Hchil 21499    +h cva 21500    .h csm 21501   0hc0v 21504   SHcsh 21508
This theorem is referenced by:  shintcl  21909  chintcli  21910  shincli  21941
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  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-fv 5263  df-ov 5861  df-sh 21786
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