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Theorem pmapj2N 37170
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 + = (+𝑃𝐾)
pmapj2.o = (⊥𝑃𝐾)
Assertion
Ref Expression
pmapj2N ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(𝑋 𝑌)) = ( ‘( ‘((𝑀𝑋) + (𝑀𝑌)))))

Proof of Theorem pmapj2N
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
2 hllat 36604 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
323ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
4 hlop 36603 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
543ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
6 simp2 1134 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
7 pmapj2.b . . . . . 6 𝐵 = (Base‘𝐾)
8 eqid 2824 . . . . . 6 (oc‘𝐾) = (oc‘𝐾)
97, 8opoccl 36435 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
105, 6, 9syl2anc 587 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
11 simp3 1135 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
127, 8opoccl 36435 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
135, 11, 12syl2anc 587 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
14 eqid 2824 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
157, 14latmcl 17662 . . . 4 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
163, 10, 13, 15syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
17 pmapj2.m . . . 4 𝑀 = (pmap‘𝐾)
18 pmapj2.o . . . 4 = (⊥𝑃𝐾)
197, 8, 17, 18polpmapN 37153 . . 3 ((𝐾 ∈ HL ∧ (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵) → ( ‘(𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))) = (𝑀‘((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
201, 16, 19syl2anc 587 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))) = (𝑀‘((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
217, 8, 17, 18polpmapN 37153 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ( ‘(𝑀𝑋)) = (𝑀‘((oc‘𝐾)‘𝑋)))
22213adant3 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑀𝑋)) = (𝑀‘((oc‘𝐾)‘𝑋)))
237, 8, 17, 18polpmapN 37153 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝐵) → ( ‘(𝑀𝑌)) = (𝑀‘((oc‘𝐾)‘𝑌)))
24233adant2 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑀𝑌)) = (𝑀‘((oc‘𝐾)‘𝑌)))
2522, 24ineq12d 4175 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑀𝑋)) ∩ ( ‘(𝑀𝑌))) = ((𝑀‘((oc‘𝐾)‘𝑋)) ∩ (𝑀‘((oc‘𝐾)‘𝑌))))
26 eqid 2824 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
277, 26, 17pmapssat 37000 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
28273adant3 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
297, 26, 17pmapssat 37000 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
30293adant2 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
31 pmapj2.p . . . . . 6 + = (+𝑃𝐾)
3226, 31, 18poldmj1N 37169 . . . . 5 ((𝐾 ∈ HL ∧ (𝑀𝑋) ⊆ (Atoms‘𝐾) ∧ (𝑀𝑌) ⊆ (Atoms‘𝐾)) → ( ‘((𝑀𝑋) + (𝑀𝑌))) = (( ‘(𝑀𝑋)) ∩ ( ‘(𝑀𝑌))))
331, 28, 30, 32syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑀𝑋) + (𝑀𝑌))) = (( ‘(𝑀𝑋)) ∩ ( ‘(𝑀𝑌))))
347, 14, 26, 17pmapmeet 37014 . . . . 5 ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌))) = ((𝑀‘((oc‘𝐾)‘𝑋)) ∩ (𝑀‘((oc‘𝐾)‘𝑌))))
351, 10, 13, 34syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌))) = ((𝑀‘((oc‘𝐾)‘𝑋)) ∩ (𝑀‘((oc‘𝐾)‘𝑌))))
3625, 33, 353eqtr4rd 2870 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌))) = ( ‘((𝑀𝑋) + (𝑀𝑌))))
3736fveq2d 6665 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑀‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))) = ( ‘( ‘((𝑀𝑋) + (𝑀𝑌)))))
38 hlol 36602 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
39 pmapj2.j . . . . 5 = (join‘𝐾)
407, 39, 14, 8oldmm4 36461 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
4138, 40syl3an1 1160 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
4241fveq2d 6665 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))) = (𝑀‘(𝑋 𝑌)))
4320, 37, 423eqtr3rd 2868 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(𝑋 𝑌)) = ( ‘( ‘((𝑀𝑋) + (𝑀𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cin 3918  wss 3919  cfv 6343  (class class class)co 7149  Basecbs 16483  occoc 16573  joincjn 17554  meetcmee 17555  Latclat 17655  OPcops 36413  OLcol 36415  Atomscatm 36504  HLchlt 36591  pmapcpmap 36738  +𝑃cpadd 37036  𝑃cpolN 37143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-riotaBAD 36194
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-undef 7935  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-psubsp 36744  df-pmap 36745  df-padd 37037  df-polarityN 37144
This theorem is referenced by:  pmapocjN  37171  pmapojoinN  37209
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