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Theorem polcon3N 29273
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
StepHypRef Expression
1 simp3 962 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  Y )
2 iinss1 3891 . . 3  |-  ( X 
C_  Y  ->  |^|_ p  e.  Y  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) ) )
3 sslin 3370 . . 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 2258 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
6 2polss.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2258 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
8 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
95, 6, 7, 8polvalN 29261 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
(  ._|_  `  Y )  =  ( A  i^i  |^|_
p  e.  Y  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1093adant3 980 . 2  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  =  ( A  i^i  |^|_ p  e.  Y  ( (
pmap `  K ) `  ( ( oc `  K ) `  p
) ) ) )
11 simp1 960 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  K  e.  HL )
12 simp2 961 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  Y  C_  A )
131, 12sstrd 3164 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  A )
145, 6, 7, 8polvalN 29261 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( A  i^i  |^|_
p  e.  X  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) ) )
1511, 13, 14syl2anc 645 . 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 3196 1  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621    i^i cin 3126    C_ wss 3127   |^|_ciin 3880   ` cfv 4673   occoc 13178   Atomscatm 28620   HLchlt 28707   pmapcpmap 28853   _|_ PcpolN 29258
This theorem is referenced by:  2polcon4bN  29274  polcon2N  29275  paddunN  29283
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-polarityN 29259
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