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Theorem dihmeetlem7N 41312
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 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ HL)
3 hlatl 39361 . . . . . 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 1194 . . . . 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 2737 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
11 dihmeetlem7.a . . . . . 6 𝐴 = (Atoms‘𝐾)
127, 8, 9, 10, 11atnle 39318 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = (0.‘𝐾)))
134, 5, 6, 12syl3anc 1373 . . . 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 39365 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ Lat)
17 simpl2 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝑋𝐵)
187, 9latmcl 18485 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1916, 17, 6, 18syl3anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
207, 8, 9latmle2 18510 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
2116, 17, 6, 20syl3anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (𝑋 𝑌) 𝑌)
22 dihmeetlem7.j . . . 4 = (join‘𝐾)
237, 8, 22, 9, 11atmod1i2 39861 . . 3 ((𝐾 ∈ HL ∧ (𝑝𝐴 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑌) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
242, 5, 19, 6, 21, 23syl131anc 1385 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → ((𝑋 𝑌) (𝑝 𝑌)) = (((𝑋 𝑌) 𝑝) 𝑌))
25 hlol 39362 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → 𝐾 ∈ OL)
277, 22, 10olj01 39226 . . 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 2785 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 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  OLcol 39175  Atomscatm 39264  AtLatcal 39265  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-psubsp 39505  df-pmap 39506  df-padd 39798
This theorem is referenced by: (None)
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