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Theorem polcon3N 28795
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 3815 . . 3  |-  ( X 
C_  Y  ->  |^|_ p  e.  Y  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) )  C_  |^|_ p  e.  X  ( ( pmap `  K ) `  ( ( oc `  K ) `  p
) ) )
3 sslin 3302 . . 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 2253 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
6 2polss.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2253 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
8 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
95, 6, 7, 8polvalN 28783 . . 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 3110 . . 3  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  X  C_  A )
145, 6, 7, 8polvalN 28783 . . 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 3142 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 3077    C_ wss 3078   |^|_ciin 3804   ` cfv 4592   occoc 13090   Atomscatm 28142   HLchlt 28229   pmapcpmap 28375   _|_ PcpolN 28780
This theorem is referenced by:  2polcon4bN  28796  polcon2N  28797  paddunN  28805
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-polarityN 28781
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