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Theorem polcon3N 33190
Description: Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polss.a  |-  A  =  ( Atoms `  K )
2polss.p  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
polcon3N  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
)

Proof of Theorem polcon3N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simp3 1007 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  Y )
2 iinss1 4315 . . 3  |-  ( X 
C_  Y  ->  |^|_ p  e.  Y  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) ) )
3 sslin 3694 . . 3  |-  ( |^|_ p  e.  Y  ( (
pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  ->  ( A  i^i  |^|_ p  e.  Y  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) 
C_  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
41, 2, 33syl 18 . 2  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  ( A  i^i  |^|_ p  e.  Y  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) 
C_  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
5 eqid 2429 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
6 2polss.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2429 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
8 2polss.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
95, 6, 7, 8polvalN 33178 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
(  ._|_  `  Y )  =  ( A  i^i  |^|_
p  e.  Y  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1093adant3 1025 . 2  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  =  ( A  i^i  |^|_ p  e.  Y  ( (
pmap `  K ) `  ( ( oc `  K ) `  p
) ) ) )
11 simp1 1005 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  K  e.  HL )
12 simp2 1006 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  Y  C_  A )
131, 12sstrd 3480 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  A )
145, 6, 7, 8polvalN 33178 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1511, 13, 14syl2anc 665 . 2  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  X )  =  ( A  i^i  |^|_ p  e.  X  ( (
pmap `  K ) `  ( ( oc `  K ) `  p
) ) ) )
164, 10, 153sstr4d 3513 1  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870    i^i cin 3441    C_ wss 3442   |^|_ciin 4303   ` cfv 5601   occoc 15160   Atomscatm 32537   HLchlt 32624   pmapcpmap 32770   _|_PcpolN 33175
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-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
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-reu 2789  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-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  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-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-polarityN 33176
This theorem is referenced by:  2polcon4bN  33191  polcon2N  33192  paddunN  33200
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