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Theorem llnmod2i2 34167
 Description: Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 ∨ 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
atmod.b 𝐵 = (Base‘𝐾)
atmod.l = (le‘𝐾)
atmod.j = (join‘𝐾)
atmod.m = (meet‘𝐾)
atmod.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
llnmod2i2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑋 (𝑃 𝑄)) 𝑌) = (𝑋 ((𝑃 𝑄) 𝑌)))

Proof of Theorem llnmod2i2
StepHypRef Expression
1 simp11 1084 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝐾 ∈ HL)
2 hllat 33668 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝐾 ∈ Lat)
4 simp13 1086 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑌𝐵)
5 simp2l 1080 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑃𝐴)
6 simp2r 1081 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑄𝐴)
7 atmod.b . . . . . 6 𝐵 = (Base‘𝐾)
8 atmod.j . . . . . 6 = (join‘𝐾)
9 atmod.a . . . . . 6 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
111, 5, 6, 10syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑃 𝑄) ∈ 𝐵)
12 simp12 1085 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑋𝐵)
13 atmod.m . . . . 5 = (meet‘𝐾)
147, 13latmcl 16875 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄) 𝑋) ∈ 𝐵)
153, 11, 12, 14syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑃 𝑄) 𝑋) ∈ 𝐵)
167, 8latjcom 16882 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((𝑃 𝑄) 𝑋) ∈ 𝐵) → (𝑌 ((𝑃 𝑄) 𝑋)) = (((𝑃 𝑄) 𝑋) 𝑌))
173, 4, 15, 16syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = (((𝑃 𝑄) 𝑋) 𝑌))
187, 8latjcl 16874 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑌 (𝑃 𝑄)) ∈ 𝐵)
193, 4, 11, 18syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑄)) ∈ 𝐵)
207, 13latmcom 16898 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 (𝑃 𝑄)) ∈ 𝐵) → (𝑋 (𝑌 (𝑃 𝑄))) = ((𝑌 (𝑃 𝑄)) 𝑋))
213, 12, 19, 20syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 (𝑌 (𝑃 𝑄))) = ((𝑌 (𝑃 𝑄)) 𝑋))
227, 8latjcom 16882 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑌𝐵) → ((𝑃 𝑄) 𝑌) = (𝑌 (𝑃 𝑄)))
233, 11, 4, 22syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑃 𝑄) 𝑌) = (𝑌 (𝑃 𝑄)))
2423oveq2d 6565 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 ((𝑃 𝑄) 𝑌)) = (𝑋 (𝑌 (𝑃 𝑄))))
25 simp3 1056 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑌 𝑋)
26 atmod.l . . . . 5 = (le‘𝐾)
277, 26, 8, 13, 9llnmod1i2 34164 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝐵𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = ((𝑌 (𝑃 𝑄)) 𝑋))
281, 4, 12, 5, 6, 25, 27syl321anc 1340 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = ((𝑌 (𝑃 𝑄)) 𝑋))
2921, 24, 283eqtr4d 2654 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 ((𝑃 𝑄) 𝑌)) = (𝑌 ((𝑃 𝑄) 𝑋)))
307, 13latmcom 16898 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑋))
313, 12, 11, 30syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑋))
3231oveq1d 6564 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑋 (𝑃 𝑄)) 𝑌) = (((𝑃 𝑄) 𝑋) 𝑌))
3317, 29, 323eqtr4rd 2655 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑋 (𝑃 𝑄)) 𝑌) = (𝑋 ((𝑃 𝑄) 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100 This theorem is referenced by:  dalawlem11  34185
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