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Theorem llnmod2i2 36991
Description: Version of modular law pmod1i 36976 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 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝐾 ∈ HL)
21hllatd 36492 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝐾 ∈ Lat)
3 simp13 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑌𝐵)
4 simp2l 1194 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑃𝐴)
5 simp2r 1195 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑄𝐴)
6 atmod.b . . . . . 6 𝐵 = (Base‘𝐾)
7 atmod.j . . . . . 6 = (join‘𝐾)
8 atmod.a . . . . . 6 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 36495 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
101, 4, 5, 9syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑃 𝑄) ∈ 𝐵)
11 simp12 1199 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑋𝐵)
12 atmod.m . . . . 5 = (meet‘𝐾)
136, 12latmcl 17654 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄) 𝑋) ∈ 𝐵)
142, 10, 11, 13syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑃 𝑄) 𝑋) ∈ 𝐵)
156, 7latjcom 17661 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((𝑃 𝑄) 𝑋) ∈ 𝐵) → (𝑌 ((𝑃 𝑄) 𝑋)) = (((𝑃 𝑄) 𝑋) 𝑌))
162, 3, 14, 15syl3anc 1366 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = (((𝑃 𝑄) 𝑋) 𝑌))
176, 7latjcl 17653 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑌 (𝑃 𝑄)) ∈ 𝐵)
182, 3, 10, 17syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑄)) ∈ 𝐵)
196, 12latmcom 17677 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 (𝑃 𝑄)) ∈ 𝐵) → (𝑋 (𝑌 (𝑃 𝑄))) = ((𝑌 (𝑃 𝑄)) 𝑋))
202, 11, 18, 19syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 (𝑌 (𝑃 𝑄))) = ((𝑌 (𝑃 𝑄)) 𝑋))
216, 7latjcom 17661 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑌𝐵) → ((𝑃 𝑄) 𝑌) = (𝑌 (𝑃 𝑄)))
222, 10, 3, 21syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑃 𝑄) 𝑌) = (𝑌 (𝑃 𝑄)))
2322oveq2d 7164 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 ((𝑃 𝑄) 𝑌)) = (𝑋 (𝑌 (𝑃 𝑄))))
24 simp3 1133 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → 𝑌 𝑋)
25 atmod.l . . . . 5 = (le‘𝐾)
266, 25, 7, 12, 8llnmod1i2 36988 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝐵𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = ((𝑌 (𝑃 𝑄)) 𝑋))
271, 3, 11, 4, 5, 24, 26syl321anc 1387 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑌 ((𝑃 𝑄) 𝑋)) = ((𝑌 (𝑃 𝑄)) 𝑋))
2820, 23, 273eqtr4d 2864 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 ((𝑃 𝑄) 𝑌)) = (𝑌 ((𝑃 𝑄) 𝑋)))
296, 12latmcom 17677 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑋))
302, 11, 10, 29syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑋))
3130oveq1d 7163 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑋 (𝑃 𝑄)) 𝑌) = (((𝑃 𝑄) 𝑋) 𝑌))
3216, 28, 313eqtr4rd 2865 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 398  w3a 1082   = wceq 1531  wcel 2108   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36391  HLchlt 36478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-psubsp 36631  df-pmap 36632  df-padd 36924
This theorem is referenced by:  dalawlem11  37009
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