Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  polcon3N Unicode version

Theorem polcon3N 30399
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 959 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  Y )
2 iinss1 4065 . . 3  |-  ( X 
C_  Y  ->  |^|_ p  e.  Y  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) ) )
3 sslin 3527 . . 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 19 . 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 2404 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
6 2polss.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2404 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
8 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
95, 6, 7, 8polvalN 30387 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
(  ._|_  `  Y )  =  ( A  i^i  |^|_
p  e.  Y  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1093adant3 977 . 2  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  =  ( A  i^i  |^|_ p  e.  Y  ( (
pmap `  K ) `  ( ( oc `  K ) `  p
) ) ) )
11 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  K  e.  HL )
12 simp2 958 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  Y  C_  A )
131, 12sstrd 3318 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  A )
145, 6, 7, 8polvalN 30387 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1511, 13, 14syl2anc 643 . 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 3351 1  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   |^|_ciin 4054   ` cfv 5413   occoc 13492   Atomscatm 29746   HLchlt 29833   pmapcpmap 29979   _|_ PcpolN 30384
This theorem is referenced by:  2polcon4bN  30400  polcon2N  30401  paddunN  30409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-polarityN 30385
  Copyright terms: Public domain W3C validator