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Theorem dihmeetlem7N 37469
 Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem7.b 𝐵 = (Base‘𝐾)
dihmeetlem7.l = (le‘𝐾)
dihmeetlem7.j = (join‘𝐾)
dihmeetlem7.m = (meet‘𝐾)
dihmeetlem7.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dihmeetlem7N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (((𝑋 𝑌) 𝑝) 𝑌) = (𝑋 𝑌))

Proof of Theorem dihmeetlem7N
StepHypRef Expression
1 simprr 763 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ¬ 𝑝 𝑌)
2 simpl1 1199 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ HL)
3 hlatl 35519 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
42, 3syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ AtLat)
5 simprl 761 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑝𝐴)
6 simpl3 1203 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑌𝐵)
7 dihmeetlem7.b . . . . . 6 𝐵 = (Base‘𝐾)
8 dihmeetlem7.l . . . . . 6 = (le‘𝐾)
9 dihmeetlem7.m . . . . . 6 = (meet‘𝐾)
10 eqid 2778 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
11 dihmeetlem7.a . . . . . 6 𝐴 = (Atoms‘𝐾)
127, 8, 9, 10, 11atnle 35476 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
134, 5, 6, 12syl3anc 1439 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
141, 13mpbid 224 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑝 𝑌) = (0.‘𝐾))
1514oveq2d 6940 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = ((𝑋 𝑌) (0.‘𝐾)))
162hllatd 35523 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ Lat)
17 simpl2 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑋𝐵)
187, 9latmcl 17442 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1916, 17, 6, 18syl3anc 1439 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
207, 8, 9latmle2 17467 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
2116, 17, 6, 20syl3anc 1439 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) 𝑌)
22 dihmeetlem7.j . . . 4 = (join‘𝐾)
237, 8, 22, 9, 11atmod1i2 36018 . . 3 ((𝐾 ∈ HL ∧ (𝑝𝐴 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑌) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
242, 5, 19, 6, 21, 23syl131anc 1451 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
25 hlol 35520 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ OL)
277, 22, 10olj01 35384 . . 3 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
2826, 19, 27syl2anc 579 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
2915, 24, 283eqtr3d 2822 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (((𝑋 𝑌) 𝑝) 𝑌) = (𝑋 𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   class class class wbr 4888  ‘cfv 6137  (class class class)co 6924  Basecbs 16259  lecple 16349  joincjn 17334  meetcmee 17335  0.cp0 17427  Latclat 17435  OLcol 35333  Atomscatm 35422  AtLatcal 35423  HLchlt 35509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-proset 17318  df-poset 17336  df-plt 17348  df-lub 17364  df-glb 17365  df-join 17366  df-meet 17367  df-p0 17429  df-lat 17436  df-clat 17498  df-oposet 35335  df-ol 35337  df-oml 35338  df-covers 35425  df-ats 35426  df-atl 35457  df-cvlat 35481  df-hlat 35510  df-psubsp 35662  df-pmap 35663  df-padd 35955 This theorem is referenced by: (None)
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