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Theorem atmod2i2 37613
Description: Version of modular law pmod2iN 37600 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
atmod2i2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑋 𝑃) 𝑌) = (𝑋 (𝑃 𝑌)))

Proof of Theorem atmod2i2
StepHypRef Expression
1 hllat 37114 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1135 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝐾 ∈ Lat)
3 simp21 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑃𝐴)
4 atmod.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 atmod.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 37040 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑃𝐵)
8 simp23 1210 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑌𝐵)
9 atmod.j . . . . . 6 = (join‘𝐾)
104, 9latjcom 17953 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) = (𝑌 𝑃))
112, 7, 8, 10syl3anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑌) = (𝑌 𝑃))
1211oveq1d 7228 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑃 𝑌) 𝑋) = ((𝑌 𝑃) 𝑋))
13 simp22 1209 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑋𝐵)
144, 9latjcl 17945 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) ∈ 𝐵)
152, 7, 8, 14syl3anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑌) ∈ 𝐵)
16 atmod.m . . . . 5 = (meet‘𝐾)
174, 16latmcom 17969 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑌) ∈ 𝐵) → (𝑋 (𝑃 𝑌)) = ((𝑃 𝑌) 𝑋))
182, 13, 15, 17syl3anc 1373 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑌)) = ((𝑃 𝑌) 𝑋))
19 simp1 1138 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝐾 ∈ HL)
20 simp3 1140 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑌 𝑋)
21 atmod.l . . . . 5 = (le‘𝐾)
224, 21, 9, 16, 5atmod1i2 37610 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑌𝐵𝑋𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑌 𝑃) 𝑋))
2319, 3, 8, 13, 20, 22syl131anc 1385 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑌 𝑃) 𝑋))
2412, 18, 233eqtr4d 2787 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑌)) = (𝑌 (𝑃 𝑋)))
254, 16latmcl 17946 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
262, 7, 13, 25syl3anc 1373 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑋) ∈ 𝐵)
274, 9latjcom 17953 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑃 𝑋) ∈ 𝐵) → (𝑌 (𝑃 𝑋)) = ((𝑃 𝑋) 𝑌))
282, 8, 26, 27syl3anc 1373 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑃 𝑋) 𝑌))
294, 16latmcom 17969 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) = (𝑋 𝑃))
302, 7, 13, 29syl3anc 1373 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑋) = (𝑋 𝑃))
3130oveq1d 7228 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑃 𝑋) 𝑌) = ((𝑋 𝑃) 𝑌))
3224, 28, 313eqtrrd 2782 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑋 𝑃) 𝑌) = (𝑋 (𝑃 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110   class class class wbr 5053  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  meetcmee 17819  Latclat 17937  Atomscatm 37014  HLchlt 37101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-psubsp 37254  df-pmap 37255  df-padd 37547
This theorem is referenced by:  llnexchb2lem  37619  dalawlem2  37623  dalawlem3  37624  dalawlem11  37632  dalawlem12  37633  cdleme15b  38026
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