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Theorem pmapj2N 35215
Description: The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapj2.b |- B = ( Base ` K )
pmapj2.j |- .\/ = ( join ` K )
pmapj2.m |- M = ( pmap ` K )
pmapj2.p |- .+ = ( +P ` K )
pmapj2.o |- ._|_ = ( _|_P ` K )
Assertion
Ref Expression
pmapj2N |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` ( X .\/ Y ) ) = ( ._|_ ` ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) ) )
Proof of Theorem pmapj2N
StepHypRef Expression
1 simp1 1061 . . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. HL )
2 hllat 34650 . . . . 5 |- ( K e. HL -> K e. Lat )
323ad2ant1 1082 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. Lat )
4 hlop 34649 . . . . . 6 |- ( K e. HL -> K e. OP )
543ad2ant1 1082 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. OP )
6 simp2 1062 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> X e. B )
7 pmapj2.b . . . . . 6 |- B = ( Base ` K )
8 eqid 2622 . . . . . 6 |- ( oc ` K ) = ( oc ` K )
97, 8opoccl 34481 . . . . 5 |- ( ( K e. OP /\ X e. B ) -> ( ( oc ` K ) ` X ) e. B )
105, 6, 9syl2anc 693 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` X ) e. B )
11 simp3 1063 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> Y e. B )
127, 8opoccl 34481 . . . . 5 |- ( ( K e. OP /\ Y e. B ) -> ( ( oc ` K ) ` Y ) e. B )
135, 11, 12syl2anc 693 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` Y ) e. B )
14 eqid 2622 . . . . 5 |- ( meet ` K ) = ( meet ` K )
157, 14latmcl 17052 . . . 4 |- ( ( K e. Lat /\ ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B ) -> ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) e. B )
163, 10, 13, 15syl3anc 1326 . . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) e. B )
17 pmapj2.m . . . 4 |- M = ( pmap ` K )
18 pmapj2.o . . . 4 |- ._|_ = ( _|_P ` K )
197, 8, 17, 18polpmapN 35198 . . 3 |- ( ( K e. HL /\ ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) e. B ) -> ( ._|_ ` ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) = ( M ` ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) )
201, 16, 19syl2anc 693 . 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) = ( M ` ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) )
217, 8, 17, 18polpmapN 35198 . . . . . 6 |- ( ( K e. HL /\ X e. B ) -> ( ._|_ ` ( M ` X ) ) = ( M ` ( ( oc ` K ) ` X ) ) )
22213adant3 1081 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( M ` X ) ) = ( M ` ( ( oc ` K ) ` X ) ) )
237, 8, 17, 18polpmapN 35198 . . . . . 6 |- ( ( K e. HL /\ Y e. B ) -> ( ._|_ ` ( M ` Y ) ) = ( M ` ( ( oc ` K ) ` Y ) ) )
24233adant2 1080 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( M ` Y ) ) = ( M ` ( ( oc ` K ) ` Y ) ) )
2522, 24ineq12d 3815 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( ._|_ ` ( M ` X ) ) i^i ( ._|_ ` ( M ` Y ) ) ) = ( ( M ` ( ( oc ` K ) ` X ) ) i^i ( M ` ( ( oc ` K ) ` Y ) ) ) )
26 eqid 2622 . . . . . . 7 |- ( Atoms ` K ) = ( Atoms ` K )
277, 26, 17pmapssat 35045 . . . . . 6 |- ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ ( Atoms ` K ) )
28273adant3 1081 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` X ) C_ ( Atoms ` K ) )
297, 26, 17pmapssat 35045 . . . . . 6 |- ( ( K e. HL /\ Y e. B ) -> ( M ` Y ) C_ ( Atoms ` K ) )
30293adant2 1080 . . . . 5 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` Y ) C_ ( Atoms ` K ) )
31 pmapj2.p . . . . . 6 |- .+ = ( +P ` K )
3226, 31, 18poldmj1N 35214 . . . . 5 |- ( ( K e. HL /\ ( M ` X ) C_ ( Atoms ` K ) /\ ( M ` Y ) C_ ( Atoms ` K ) ) -> ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) = ( ( ._|_ ` ( M ` X ) ) i^i ( ._|_ ` ( M ` Y ) ) ) )
331, 28, 30, 32syl3anc 1326 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) = ( ( ._|_ ` ( M ` X ) ) i^i ( ._|_ ` ( M ` Y ) ) ) )
347, 14, 26, 17pmapmeet 35059 . . . . 5 |- ( ( K e. HL /\ ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B ) -> ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) = ( ( M ` ( ( oc ` K ) ` X ) ) i^i ( M ` ( ( oc ` K ) ` Y ) ) ) )
351, 10, 13, 34syl3anc 1326 . . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) = ( ( M ` ( ( oc ` K ) ` X ) ) i^i ( M ` ( ( oc ` K ) ` Y ) ) ) )
3625, 33, 353eqtr4rd 2667 . . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) = ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) )
3736fveq2d 6195 . 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( M ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) = ( ._|_ ` ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) ) )
38 hlol 34648 . . . 4 |- ( K e. HL -> K e. OL )
39 pmapj2.j . . . . 5 |- .\/ = ( join ` K )
407, 39, 14, 8oldmm4 34507 . . . 4 |- ( ( K e. OL /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) = ( X .\/ Y ) )
4138, 40syl3an1 1359 . . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) = ( X .\/ Y ) )
4241fveq2d 6195 . 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) ) = ( M ` ( X .\/ Y ) ) )
4320, 37, 423eqtr3rd 2665 1 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( M ` ( X .\/ Y ) ) = ( ._|_ ` ( ._|_ ` ( ( M ` X ) .+ ( M ` Y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   occoc 15949   joincjn 16944   meetcmee 16945   Latclat 17045   OPcops 34459   OLcol 34461   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   +Pcpadd 35081   _|_PcpolN 35188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189
This theorem is referenced by:  pmapocjN  35216  pmapojoinN  35254
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