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Theorem shintcli 26817
Description: Closure of intersection of a nonempty 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 463 . . . 4  |-  A  =/=  (/)
3 n0 3777 . . . . 5  |-  ( A  =/=  (/)  <->  E. z  z  e.  A )
4 intss1 4273 . . . . . . 7  |-  ( z  e.  A  ->  |^| A  C_  z )
51simpli 459 . . . . . . . . 9  |-  A  C_  SH
65sseli 3466 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  SH )
7 shss 26698 . . . . . . . 8  |-  ( z  e.  SH  ->  z  C_ 
~H )
86, 7syl 17 . . . . . . 7  |-  ( z  e.  A  ->  z  C_ 
~H )
94, 8sstrd 3480 . . . . . 6  |-  ( z  e.  A  ->  |^| A  C_ 
~H )
109exlimiv 1769 . . . . 5  |-  ( E. z  z  e.  A  ->  |^| A  C_  ~H )
113, 10sylbi 198 . . . 4  |-  ( A  =/=  (/)  ->  |^| A  C_  ~H )
122, 11ax-mp 5 . . 3  |-  |^| A  C_ 
~H
13 ax-hv0cl 26491 . . . . . 6  |-  0h  e.  ~H
1413elexi 3097 . . . . 5  |-  0h  e.  _V
1514elint2 4265 . . . 4  |-  ( 0h  e.  |^| A  <->  A. z  e.  A  0h  e.  z )
16 sh0 26704 . . . . 5  |-  ( z  e.  SH  ->  0h  e.  z )
176, 16syl 17 . . . 4  |-  ( z  e.  A  ->  0h  e.  z )
1815, 17mprgbir 2796 . . 3  |-  0h  e.  |^| A
1912, 18pm3.2i 456 . 2  |-  ( |^| A  C_  ~H  /\  0h  e.  |^| A )
20 elinti 4267 . . . . . . . . 9  |-  ( x  e.  |^| A  ->  (
z  e.  A  ->  x  e.  z )
)
2120com12 32 . . . . . . . 8  |-  ( z  e.  A  ->  (
x  e.  |^| A  ->  x  e.  z ) )
22 elinti 4267 . . . . . . . . 9  |-  ( y  e.  |^| A  ->  (
z  e.  A  -> 
y  e.  z ) )
2322com12 32 . . . . . . . 8  |-  ( z  e.  A  ->  (
y  e.  |^| A  ->  y  e.  z ) )
24 shaddcl 26705 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
256, 24syl3an1 1297 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
26253expib 1208 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  +h  y )  e.  z ) )
2721, 23, 26syl2and 485 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  |^| A  /\  y  e.  |^| A )  ->  (
x  +h  y )  e.  z ) )
2827com12 32 . . . . . 6  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( z  e.  A  ->  ( x  +h  y )  e.  z ) )
2928ralrimiv 2844 . . . . 5  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  A. z  e.  A  ( x  +h  y )  e.  z )
30 ovex 6333 . . . . . 6  |-  ( x  +h  y )  e. 
_V
3130elint2 4265 . . . . 5  |-  ( ( x  +h  y )  e.  |^| A  <->  A. z  e.  A  ( x  +h  y )  e.  z )
3229, 31sylibr 215 . . . 4  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( x  +h  y )  e.  |^| A )
3332rgen2a 2859 . . 3  |-  A. x  e.  |^| A A. y  e.  |^| A ( x  +h  y )  e. 
|^| A
34 shmulcl 26706 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
356, 34syl3an1 1297 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
36353expib 1208 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  z )  ->  ( x  .h  y )  e.  z ) )
3723, 36sylan2d 484 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  |^| A
)  ->  ( x  .h  y )  e.  z ) )
3837com12 32 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( z  e.  A  ->  ( x  .h  y )  e.  z ) )
3938ralrimiv 2844 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  A. z  e.  A  ( x  .h  y
)  e.  z )
40 ovex 6333 . . . . . 6  |-  ( x  .h  y )  e. 
_V
4140elint2 4265 . . . . 5  |-  ( ( x  .h  y )  e.  |^| A  <->  A. z  e.  A  ( x  .h  y )  e.  z )
4239, 41sylibr 215 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( x  .h  y )  e.  |^| A )
4342rgen2 2857 . . 3  |-  A. x  e.  CC  A. y  e. 
|^| A ( x  .h  y )  e. 
|^| A
4433, 43pm3.2i 456 . 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 26697 . 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 928 1  |-  |^| A  e.  SH
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370   E.wex 1659    e. wcel 1870    =/= wne 2625   A.wral 2782    C_ wss 3442   (/)c0 3767   |^|cint 4258  (class class class)co 6305   CCcc 9536   ~Hchil 26407    +h cva 26408    .h csm 26409   0hc0v 26412   SHcsh 26416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-hilex 26487  ax-hfvadd 26488  ax-hv0cl 26491  ax-hfvmul 26493
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-sh 26695
This theorem is referenced by:  shintcl  26818  chintcli  26819  shincli  26850
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