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Theorem atmod2i1 37115
Description: Version of modular law pmod2iN 37103 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-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
atmod2i1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑋 𝑌) 𝑃) = (𝑋 (𝑌 𝑃)))

Proof of Theorem atmod2i1
StepHypRef Expression
1 hllat 36617 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝐾 ∈ Lat)
3 simp22 1204 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑋𝐵)
4 simp23 1205 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑌𝐵)
5 atmod.b . . . . 5 𝐵 = (Base‘𝐾)
6 atmod.m . . . . 5 = (meet‘𝐾)
75, 6latmcom 17676 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
82, 3, 4, 7syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
98oveq2d 7156 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑋 𝑌)) = (𝑃 (𝑌 𝑋)))
10 simp21 1203 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃𝐴)
11 atmod.a . . . . 5 𝐴 = (Atoms‘𝐾)
125, 11atbase 36543 . . . 4 (𝑃𝐴𝑃𝐵)
1310, 12syl 17 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃𝐵)
145, 6latmcl 17653 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
152, 3, 4, 14syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 𝑌) ∈ 𝐵)
16 atmod.j . . . 4 = (join‘𝐾)
175, 16latjcom 17660 . . 3 ((𝐾 ∈ Lat ∧ 𝑃𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑃 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑃))
182, 13, 15, 17syl3anc 1368 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑃))
195, 16latjcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) ∈ 𝐵)
202, 13, 4, 19syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 𝑌) ∈ 𝐵)
215, 6latmcom 17676 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑌) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑌) 𝑋) = (𝑋 (𝑃 𝑌)))
222, 20, 3, 21syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑃 𝑌) 𝑋) = (𝑋 (𝑃 𝑌)))
23 simp1 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝐾 ∈ HL)
24 simp3 1135 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃 𝑋)
25 atmod.l . . . . 5 = (le‘𝐾)
265, 25, 16, 6, 11atmod1i1 37111 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑌𝐵𝑋𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = ((𝑃 𝑌) 𝑋))
2723, 10, 4, 3, 24, 26syl131anc 1380 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = ((𝑃 𝑌) 𝑋))
285, 16latjcom 17660 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑃𝐵) → (𝑌 𝑃) = (𝑃 𝑌))
292, 4, 13, 28syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑌 𝑃) = (𝑃 𝑌))
3029oveq2d 7156 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 (𝑌 𝑃)) = (𝑋 (𝑃 𝑌)))
3122, 27, 303eqtr4d 2867 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = (𝑋 (𝑌 𝑃)))
329, 18, 313eqtr3d 2865 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑋 𝑌) 𝑃) = (𝑋 (𝑌 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Latclat 17646  Atomscatm 36517  HLchlt 36604
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-lat 17647  df-clat 17709  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-psubsp 36757  df-pmap 36758  df-padd 37050
This theorem is referenced by:  lhpmod6i1  37293  trljat1  37420  trljat2  37421  cdlemc1  37445  cdlemc6  37450  cdleme16b  37533  cdleme20c  37565  cdleme20j  37572  cdleme22e  37598  cdleme22eALTN  37599  cdlemkid1  38176  dihmeetlem5  38562
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