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Theorem pmapj2N 38795
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 1136 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ HL)
2 hllat 38228 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
323ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
4 hlop 38227 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
543ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
6 simp2 1137 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
7 pmapj2.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
8 eqid 2732 . . . . . 6 (ocβ€˜πΎ) = (ocβ€˜πΎ)
97, 8opoccl 38059 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
105, 6, 9syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
11 simp3 1138 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
127, 8opoccl 38059 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
135, 11, 12syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
14 eqid 2732 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
157, 14latmcl 18392 . . . 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 1371 . . 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 38778 . . 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 584 . 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 38778 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘‹)))
22213adant3 1132 . . . . 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 38778 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(π‘€β€˜π‘Œ)) = (π‘€β€˜((ocβ€˜πΎ)β€˜π‘Œ)))
24233adant2 1131 . . . . 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 4213 . . . 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 2732 . . . . . . 7 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
277, 26, 17pmapssat 38625 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
28273adant3 1132 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
297, 26, 17pmapssat 38625 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) βŠ† (Atomsβ€˜πΎ))
30293adant2 1131 . . . . 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 38794 . . . . 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 1371 . . . 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 38639 . . . . 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 1371 . . . 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 2783 . . 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 6895 . 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 38226 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
39 pmapj2.j . . . . 5 ∨ = (joinβ€˜πΎ)
407, 39, 14, 8oldmm4 38085 . . . 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 1163 . . 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 6895 . 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 2781 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 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  occoc 17204  joincjn 18263  meetcmee 18264  Latclat 18383  OPcops 38037  OLcol 38039  Atomscatm 38128  HLchlt 38215  pmapcpmap 38363  +𝑃cpadd 38661  βŠ₯𝑃cpolN 38768
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-polarityN 38769
This theorem is referenced by:  pmapocjN  38796  pmapojoinN  38834
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