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Theorem atmod2i1 35650
Description: Version of modular law pmod2iN 35638 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 35153 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1128 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝐾 ∈ Lat)
3 simp22 1250 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑋𝐵)
4 simp23 1251 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑌𝐵)
5 atmod.b . . . . 5 𝐵 = (Base‘𝐾)
6 atmod.m . . . . 5 = (meet‘𝐾)
75, 6latmcom 17276 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
82, 3, 4, 7syl3anc 1477 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
98oveq2d 6829 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑋 𝑌)) = (𝑃 (𝑌 𝑋)))
10 simp21 1249 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃𝐴)
11 atmod.a . . . . 5 𝐴 = (Atoms‘𝐾)
125, 11atbase 35079 . . . 4 (𝑃𝐴𝑃𝐵)
1310, 12syl 17 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃𝐵)
145, 6latmcl 17253 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
152, 3, 4, 14syl3anc 1477 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 𝑌) ∈ 𝐵)
16 atmod.j . . . 4 = (join‘𝐾)
175, 16latjcom 17260 . . 3 ((𝐾 ∈ Lat ∧ 𝑃𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑃 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑃))
182, 13, 15, 17syl3anc 1477 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑃))
195, 16latjcl 17252 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) ∈ 𝐵)
202, 13, 4, 19syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 𝑌) ∈ 𝐵)
215, 6latmcom 17276 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑌) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑌) 𝑋) = (𝑋 (𝑃 𝑌)))
222, 20, 3, 21syl3anc 1477 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑃 𝑌) 𝑋) = (𝑋 (𝑃 𝑌)))
23 simp1 1131 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝐾 ∈ HL)
24 simp3 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → 𝑃 𝑋)
25 atmod.l . . . . 5 = (le‘𝐾)
265, 25, 16, 6, 11atmod1i1 35646 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑌𝐵𝑋𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = ((𝑃 𝑌) 𝑋))
2723, 10, 4, 3, 24, 26syl131anc 1490 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = ((𝑃 𝑌) 𝑋))
285, 16latjcom 17260 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑃𝐵) → (𝑌 𝑃) = (𝑃 𝑌))
292, 4, 13, 28syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑌 𝑃) = (𝑃 𝑌))
3029oveq2d 6829 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑋 (𝑌 𝑃)) = (𝑋 (𝑃 𝑌)))
3122, 27, 303eqtr4d 2804 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑌 𝑋)) = (𝑋 (𝑌 𝑃)))
329, 18, 313eqtr3d 2802 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑋 𝑌) 𝑃) = (𝑋 (𝑌 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35053  HLchlt 35140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-psubsp 35292  df-pmap 35293  df-padd 35585
This theorem is referenced by:  lhpmod6i1  35828  trljat1  35956  trljat2  35957  cdlemc1  35981  cdlemc6  35986  cdleme16b  36069  cdleme20c  36101  cdleme20j  36108  cdleme22e  36134  cdleme22eALTN  36135  cdlemkid1  36712  dihmeetlem5  37099
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