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Theorem dihmeetlem7N 41293
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 773 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ¬ 𝑝 𝑌)
2 simpl1 1190 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ HL)
3 hlatl 39342 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
42, 3syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ AtLat)
5 simprl 771 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑝𝐴)
6 simpl3 1192 . . . . 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 2735 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
11 dihmeetlem7.a . . . . . 6 𝐴 = (Atoms‘𝐾)
127, 8, 9, 10, 11atnle 39299 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
134, 5, 6, 12syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
141, 13mpbid 232 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑝 𝑌) = (0.‘𝐾))
1514oveq2d 7447 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = ((𝑋 𝑌) (0.‘𝐾)))
162hllatd 39346 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ Lat)
17 simpl2 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑋𝐵)
187, 9latmcl 18498 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1916, 17, 6, 18syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
207, 8, 9latmle2 18523 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
2116, 17, 6, 20syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) 𝑌)
22 dihmeetlem7.j . . . 4 = (join‘𝐾)
237, 8, 22, 9, 11atmod1i2 39842 . . 3 ((𝐾 ∈ HL ∧ (𝑝𝐴 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑌) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
242, 5, 19, 6, 21, 23syl131anc 1382 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
25 hlol 39343 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ OL)
277, 22, 10olj01 39207 . . 3 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
2826, 19, 27syl2anc 584 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (0.‘𝐾)) = (𝑋 𝑌))
2915, 24, 283eqtr3d 2783 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  0.cp0 18481  Latclat 18489  OLcol 39156  Atomscatm 39245  AtLatcal 39246  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779
This theorem is referenced by: (None)
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