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Theorem pmapjlln1 39239
Description: The projective map of the join of a lattice element and a lattice line (expressed as the join 𝑄 ∨ 𝑅 of two atoms). (Contributed by NM, 16-Sep-2012.)
Hypotheses
Ref Expression
pmapjat.b 𝐡 = (Baseβ€˜πΎ)
pmapjat.j ∨ = (joinβ€˜πΎ)
pmapjat.a 𝐴 = (Atomsβ€˜πΎ)
pmapjat.m 𝑀 = (pmapβ€˜πΎ)
pmapjat.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmapjlln1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((π‘€β€˜π‘‹) + (π‘€β€˜(𝑄 ∨ 𝑅))))

Proof of Theorem pmapjlln1
StepHypRef Expression
1 simpl 482 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
2 pmapjat.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 pmapjat.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
4 pmapjat.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
52, 3, 4pmapssat 39143 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) βŠ† 𝐴)
653ad2antr1 1185 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜π‘‹) βŠ† 𝐴)
7 simpr2 1192 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
82, 3atbase 38672 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
97, 8syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
102, 3, 4pmapssat 39143 . . . 4 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐡) β†’ (π‘€β€˜π‘„) βŠ† 𝐴)
119, 10syldan 590 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜π‘„) βŠ† 𝐴)
12 simpr3 1193 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
132, 3atbase 38672 . . . . 5 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ 𝐡)
1412, 13syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐡)
152, 3, 4pmapssat 39143 . . . 4 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐡) β†’ (π‘€β€˜π‘…) βŠ† 𝐴)
1614, 15syldan 590 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜π‘…) βŠ† 𝐴)
17 pmapjat.p . . . 4 + = (+π‘ƒβ€˜πΎ)
183, 17paddass 39222 . . 3 ((𝐾 ∈ HL ∧ ((π‘€β€˜π‘‹) βŠ† 𝐴 ∧ (π‘€β€˜π‘„) βŠ† 𝐴 ∧ (π‘€β€˜π‘…) βŠ† 𝐴)) β†’ (((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)) + (π‘€β€˜π‘…)) = ((π‘€β€˜π‘‹) + ((π‘€β€˜π‘„) + (π‘€β€˜π‘…))))
191, 6, 11, 16, 18syl13anc 1369 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)) + (π‘€β€˜π‘…)) = ((π‘€β€˜π‘‹) + ((π‘€β€˜π‘„) + (π‘€β€˜π‘…))))
20 hllat 38746 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2120adantr 480 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
22 simpr1 1191 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
23 pmapjat.j . . . . . 6 ∨ = (joinβ€˜πΎ)
242, 23latjcl 18404 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
2521, 22, 9, 24syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
262, 23, 3, 4, 17pmapjat1 39237 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑄) ∈ 𝐡 ∧ 𝑅 ∈ 𝐴) β†’ (π‘€β€˜((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((π‘€β€˜(𝑋 ∨ 𝑄)) + (π‘€β€˜π‘…)))
271, 25, 12, 26syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((π‘€β€˜(𝑋 ∨ 𝑄)) + (π‘€β€˜π‘…)))
282, 23latjass 18448 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑅 ∈ 𝐡)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
2921, 22, 9, 14, 28syl13anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
3029fveq2d 6889 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜((𝑋 ∨ 𝑄) ∨ 𝑅)) = (π‘€β€˜(𝑋 ∨ (𝑄 ∨ 𝑅))))
312, 23, 3, 4, 17pmapjat1 39237 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (π‘€β€˜(𝑋 ∨ 𝑄)) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)))
32313adant3r3 1181 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑋 ∨ 𝑄)) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)))
3332oveq1d 7420 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((π‘€β€˜(𝑋 ∨ 𝑄)) + (π‘€β€˜π‘…)) = (((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)) + (π‘€β€˜π‘…)))
3427, 30, 333eqtr3d 2774 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑋 ∨ (𝑄 ∨ 𝑅))) = (((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)) + (π‘€β€˜π‘…)))
352, 23, 3, 4, 17pmapjat1 39237 . . . 4 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐡 ∧ 𝑅 ∈ 𝐴) β†’ (π‘€β€˜(𝑄 ∨ 𝑅)) = ((π‘€β€˜π‘„) + (π‘€β€˜π‘…)))
361, 9, 12, 35syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑄 ∨ 𝑅)) = ((π‘€β€˜π‘„) + (π‘€β€˜π‘…)))
3736oveq2d 7421 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((π‘€β€˜π‘‹) + (π‘€β€˜(𝑄 ∨ 𝑅))) = ((π‘€β€˜π‘‹) + ((π‘€β€˜π‘„) + (π‘€β€˜π‘…))))
3819, 34, 373eqtr4d 2776 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((π‘€β€˜π‘‹) + (π‘€β€˜(𝑄 ∨ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  joincjn 18276  Latclat 18396  Atomscatm 38646  HLchlt 38733  pmapcpmap 38881  +𝑃cpadd 39179
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-pmap 38888  df-padd 39180
This theorem is referenced by:  llnmod1i2  39244
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