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Theorem shintcli 22833
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 450 . . . 4  |-  A  =/=  (/)
3 n0 3639 . . . . 5  |-  ( A  =/=  (/)  <->  E. z  z  e.  A )
4 intss1 4067 . . . . . . 7  |-  ( z  e.  A  ->  |^| A  C_  z )
51simpli 446 . . . . . . . . 9  |-  A  C_  SH
65sseli 3346 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  SH )
7 shss 22714 . . . . . . . 8  |-  ( z  e.  SH  ->  z  C_ 
~H )
86, 7syl 16 . . . . . . 7  |-  ( z  e.  A  ->  z  C_ 
~H )
94, 8sstrd 3360 . . . . . 6  |-  ( z  e.  A  ->  |^| A  C_ 
~H )
109exlimiv 1645 . . . . 5  |-  ( E. z  z  e.  A  ->  |^| A  C_  ~H )
113, 10sylbi 189 . . . 4  |-  ( A  =/=  (/)  ->  |^| A  C_  ~H )
122, 11ax-mp 8 . . 3  |-  |^| A  C_ 
~H
13 ax-hv0cl 22508 . . . . . 6  |-  0h  e.  ~H
1413elexi 2967 . . . . 5  |-  0h  e.  _V
1514elint2 4059 . . . 4  |-  ( 0h  e.  |^| A  <->  A. z  e.  A  0h  e.  z )
16 sh0 22720 . . . . 5  |-  ( z  e.  SH  ->  0h  e.  z )
176, 16syl 16 . . . 4  |-  ( z  e.  A  ->  0h  e.  z )
1815, 17mprgbir 2778 . . 3  |-  0h  e.  |^| A
1912, 18pm3.2i 443 . 2  |-  ( |^| A  C_  ~H  /\  0h  e.  |^| A )
20 elinti 4061 . . . . . . . . 9  |-  ( x  e.  |^| A  ->  (
z  e.  A  ->  x  e.  z )
)
2120com12 30 . . . . . . . 8  |-  ( z  e.  A  ->  (
x  e.  |^| A  ->  x  e.  z ) )
22 elinti 4061 . . . . . . . . 9  |-  ( y  e.  |^| A  ->  (
z  e.  A  -> 
y  e.  z ) )
2322com12 30 . . . . . . . 8  |-  ( z  e.  A  ->  (
y  e.  |^| A  ->  y  e.  z ) )
24 shaddcl 22721 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
256, 24syl3an1 1218 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  z  /\  y  e.  z )  ->  ( x  +h  y
)  e.  z )
26253expib 1157 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  +h  y )  e.  z ) )
2721, 23, 26syl2and 471 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  |^| A  /\  y  e.  |^| A )  ->  (
x  +h  y )  e.  z ) )
2827com12 30 . . . . . 6  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( z  e.  A  ->  ( x  +h  y )  e.  z ) )
2928ralrimiv 2790 . . . . 5  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  A. z  e.  A  ( x  +h  y )  e.  z )
30 ovex 6108 . . . . . 6  |-  ( x  +h  y )  e. 
_V
3130elint2 4059 . . . . 5  |-  ( ( x  +h  y )  e.  |^| A  <->  A. z  e.  A  ( x  +h  y )  e.  z )
3229, 31sylibr 205 . . . 4  |-  ( ( x  e.  |^| A  /\  y  e.  |^| A
)  ->  ( x  +h  y )  e.  |^| A )
3332rgen2a 2774 . . 3  |-  A. x  e.  |^| A A. y  e.  |^| A ( x  +h  y )  e. 
|^| A
34 shmulcl 22722 . . . . . . . . . 10  |-  ( ( z  e.  SH  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
356, 34syl3an1 1218 . . . . . . . . 9  |-  ( ( z  e.  A  /\  x  e.  CC  /\  y  e.  z )  ->  (
x  .h  y )  e.  z )
36353expib 1157 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  z )  ->  ( x  .h  y )  e.  z ) )
3723, 36sylan2d 470 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  CC  /\  y  e.  |^| A
)  ->  ( x  .h  y )  e.  z ) )
3837com12 30 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( z  e.  A  ->  ( x  .h  y )  e.  z ) )
3938ralrimiv 2790 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  A. z  e.  A  ( x  .h  y
)  e.  z )
40 ovex 6108 . . . . . 6  |-  ( x  .h  y )  e. 
_V
4140elint2 4059 . . . . 5  |-  ( ( x  .h  y )  e.  |^| A  <->  A. z  e.  A  ( x  .h  y )  e.  z )
4239, 41sylibr 205 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  |^| A )  ->  ( x  .h  y )  e.  |^| A )
4342rgen2 2804 . . 3  |-  A. x  e.  CC  A. y  e. 
|^| A ( x  .h  y )  e. 
|^| A
4433, 43pm3.2i 443 . 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 22713 . 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 888 1  |-  |^| A  e.  SH
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   |^|cint 4052  (class class class)co 6083   CCcc 8990   ~Hchil 22424    +h cva 22425    .h csm 22426   0hc0v 22429   SHcsh 22433
This theorem is referenced by:  shintcl  22834  chintcli  22835  shincli  22866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-hilex 22504  ax-hfvadd 22505  ax-hv0cl 22508  ax-hfvmul 22510
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-sh 22711
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