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Theorem atmod2i2 35670
Description: Version of modular law pmod2iN 35657 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 35172 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1127 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝐾 ∈ Lat)
3 simp21 1248 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑃𝐴)
4 atmod.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 atmod.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 35098 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑃𝐵)
8 simp23 1250 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑌𝐵)
9 atmod.j . . . . . 6 = (join‘𝐾)
104, 9latjcom 17267 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) = (𝑌 𝑃))
112, 7, 8, 10syl3anc 1476 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑌) = (𝑌 𝑃))
1211oveq1d 6808 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑃 𝑌) 𝑋) = ((𝑌 𝑃) 𝑋))
13 simp22 1249 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑋𝐵)
144, 9latjcl 17259 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵) → (𝑃 𝑌) ∈ 𝐵)
152, 7, 8, 14syl3anc 1476 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑌) ∈ 𝐵)
16 atmod.m . . . . 5 = (meet‘𝐾)
174, 16latmcom 17283 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑌) ∈ 𝐵) → (𝑋 (𝑃 𝑌)) = ((𝑃 𝑌) 𝑋))
182, 13, 15, 17syl3anc 1476 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑌)) = ((𝑃 𝑌) 𝑋))
19 simp1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝐾 ∈ HL)
20 simp3 1132 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → 𝑌 𝑋)
21 atmod.l . . . . 5 = (le‘𝐾)
224, 21, 9, 16, 5atmod1i2 35667 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑌𝐵𝑋𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑌 𝑃) 𝑋))
2319, 3, 8, 13, 20, 22syl131anc 1489 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑌 𝑃) 𝑋))
2412, 18, 233eqtr4d 2815 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑋 (𝑃 𝑌)) = (𝑌 (𝑃 𝑋)))
254, 16latmcl 17260 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
262, 7, 13, 25syl3anc 1476 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑋) ∈ 𝐵)
274, 9latjcom 17267 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑃 𝑋) ∈ 𝐵) → (𝑌 (𝑃 𝑋)) = ((𝑃 𝑋) 𝑌))
282, 8, 26, 27syl3anc 1476 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑌 (𝑃 𝑋)) = ((𝑃 𝑋) 𝑌))
294, 16latmcom 17283 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) = (𝑋 𝑃))
302, 7, 13, 29syl3anc 1476 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → (𝑃 𝑋) = (𝑋 𝑃))
3130oveq1d 6808 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑃 𝑋) 𝑌) = ((𝑋 𝑃) 𝑌))
3224, 28, 313eqtrrd 2810 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑋 𝑃) 𝑌) = (𝑋 (𝑃 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  HLchlt 35159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-psubsp 35311  df-pmap 35312  df-padd 35604
This theorem is referenced by:  llnexchb2lem  35676  dalawlem2  35680  dalawlem3  35681  dalawlem11  35689  dalawlem12  35690  cdleme15b  36084
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