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Theorem pmapj2N 39434
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 𝐡 = (Baseβ€˜πΎ)
pmapj2.j ∨ = (joinβ€˜πΎ)
pmapj2.m 𝑀 = (pmapβ€˜πΎ)
pmapj2.p + = (+π‘ƒβ€˜πΎ)
pmapj2.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmapj2N ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ( βŠ₯ β€˜( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))))
Proof of Theorem pmapj2N
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ HL)
2 hllat 38867 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
323ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
4 hlop 38866 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
543ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
6 simp2 1134 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
7 pmapj2.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
8 eqid 2728 . . . . . 6 (ocβ€˜πΎ) = (ocβ€˜πΎ)
97, 8opoccl 38698 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
105, 6, 9syl2anc 582 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
11 simp3 1135 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
127, 8opoccl 38698 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
135, 11, 12syl2anc 582 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
14 eqid 2728 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
157, 14latmcl 18439 . . . 4 ((𝐾 ∈ Lat ∧ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)) ∈ 𝐡)
163, 10, 13, 15syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)) ∈ 𝐡)
17 pmapj2.m . . . 4 𝑀 = (pmapβ€˜πΎ)
18 pmapj2.o . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
197, 8, 17, 18polpmapN 39417 . . 3 ((𝐾 ∈ HL ∧ (((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)) ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))) = (π‘€β€˜((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))))
201, 16, 19syl2anc 582 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))) = (π‘€β€˜((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))))
217, 8, 17, 18polpmapN 39417 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)))
22213adant3 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)))
237, 8, 17, 18polpmapN 39417 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘Œ)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ)))
24233adant2 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘Œ)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ)))
2522, 24ineq12d 4215 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜(π‘€β€˜π‘‹)) ∩ ( βŠ₯ β€˜(π‘€β€˜π‘Œ))) = ((π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)) ∩ (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ))))
26 eqid 2728 . . . . . . 7 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
277, 26, 17pmapssat 39264 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
28273adant3 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
297, 26, 17pmapssat 39264 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) βŠ† (Atomsβ€˜πΎ))
30293adant2 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) βŠ† (Atomsβ€˜πΎ))
31 pmapj2.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
3226, 31, 18poldmj1N 39433 . . . . 5 ((𝐾 ∈ HL ∧ (π‘€β€˜π‘‹) βŠ† (Atomsβ€˜πΎ) ∧ (π‘€β€˜π‘Œ) βŠ† (Atomsβ€˜πΎ)) β†’ ( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ))) = (( βŠ₯ β€˜(π‘€β€˜π‘‹)) ∩ ( βŠ₯ β€˜(π‘€β€˜π‘Œ))))
331, 28, 30, 32syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ))) = (( βŠ₯ β€˜(π‘€β€˜π‘‹)) ∩ ( βŠ₯ β€˜(π‘€β€˜π‘Œ))))
347, 14, 26, 17pmapmeet 39278 . . . . 5 ((𝐾 ∈ HL ∧ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡) β†’ (π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ))) = ((π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)) ∩ (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ))))
351, 10, 13, 34syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ))) = ((π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)) ∩ (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ))))
3625, 33, 353eqtr4rd 2779 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ))) = ( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ))))
3736fveq2d 6906 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))) = ( βŠ₯ β€˜( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))))
38 hlol 38865 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
39 pmapj2.j . . . . 5 ∨ = (joinβ€˜πΎ)
407, 39, 14, 8oldmm4 38724 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ))) = (𝑋 ∨ π‘Œ))
4138, 40syl3an1 1160 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ))) = (𝑋 ∨ π‘Œ))
4241fveq2d 6906 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Œ)))) = (π‘€β€˜(𝑋 ∨ π‘Œ)))
4320, 37, 423eqtr3rd 2777 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ( βŠ₯ β€˜( βŠ₯ β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3948   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  occoc 17248  joincjn 18310  meetcmee 18311  Latclat 18430  OPcops 38676  OLcol 38678  Atomscatm 38767  HLchlt 38854  pmapcpmap 39002  +𝑃cpadd 39300  βŠ₯𝑃cpolN 39407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-polarityN 39408
This theorem is referenced by:  pmapocjN  39435  pmapojoinN  39473
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