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Theorem dihmeetlem7N 35411
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 792 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ¬ 𝑝 𝑌)
2 simpl1 1057 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ HL)
3 hlatl 33459 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
42, 3syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ AtLat)
5 simprl 790 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑝𝐴)
6 simpl3 1059 . . . . 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 2610 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
11 dihmeetlem7.a . . . . . 6 𝐴 = (Atoms‘𝐾)
127, 8, 9, 10, 11atnle 33416 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
134, 5, 6, 12syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
141, 13mpbid 221 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑝 𝑌) = (0.‘𝐾))
1514oveq2d 6543 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = ((𝑋 𝑌) (0.‘𝐾)))
16 hllat 33462 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
172, 16syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ Lat)
18 simpl2 1058 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑋𝐵)
197, 9latmcl 16824 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2017, 18, 6, 19syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
217, 8, 9latmle2 16849 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
2217, 18, 6, 21syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) 𝑌)
23 dihmeetlem7.j . . . 4 = (join‘𝐾)
247, 8, 23, 9, 11atmod1i2 33957 . . 3 ((𝐾 ∈ HL ∧ (𝑝𝐴 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑌) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
252, 5, 20, 6, 22, 24syl131anc 1331 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
26 hlol 33460 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
272, 26syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ OL)
287, 23, 10olj01 33324 . . 3 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
2927, 20, 28syl2anc 691 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
3015, 25, 293eqtr3d 2652 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  meetcmee 16717  0.cp0 16809  Latclat 16817  OLcol 33273  Atomscatm 33362  AtLatcal 33363  HLchlt 33449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-iin 4453  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-lat 16818  df-clat 16880  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-psubsp 33601  df-pmap 33602  df-padd 33894
This theorem is referenced by: (None)
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