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Theorem polcon3N 29374
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 )
)
Dummy variable  p is distinct from all other variables.

Proof of Theorem polcon3N
StepHypRef Expression
1 simp3 959 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  Y )
2 iinss1 3919 . . 3  |-  ( X 
C_  Y  ->  |^|_ p  e.  Y  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) ) )
3 sslin 3397 . . 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 20 . 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 2285 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
6 2polss.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2285 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
8 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
95, 6, 7, 8polvalN 29362 . . 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 3191 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  A )
145, 6, 7, 8polvalN 29362 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1511, 13, 14syl2anc 644 . 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 3223 1  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 936    = wceq 1624    e. wcel 1685    i^i cin 3153    C_ wss 3154   |^|_ciin 3908   ` cfv 5222   occoc 13211   Atomscatm 28721   HLchlt 28808   pmapcpmap 28954   _|_ PcpolN 29359
This theorem is referenced by:  2polcon4bN  29375  polcon2N  29376  paddunN  29384
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-polarityN 29360
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