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Theorem dihmeetlem6 40168
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihmeetlem6.b 𝐵 = (Base‘𝐾)
dihmeetlem6.l = (le‘𝐾)
dihmeetlem6.h 𝐻 = (LHyp‘𝐾)
dihmeetlem6.j = (join‘𝐾)
dihmeetlem6.m = (meet‘𝐾)
dihmeetlem6.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dihmeetlem6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ¬ (𝑋 (𝑌 𝑄)) 𝑊)

Proof of Theorem dihmeetlem6
StepHypRef Expression
1 simprlr 778 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ¬ 𝑄 𝑊)
2 simpl1l 1224 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝐾 ∈ HL)
32hllatd 38222 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝐾 ∈ Lat)
4 simpl2 1192 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑋𝐵)
5 simpl3 1193 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑌𝐵)
6 dihmeetlem6.b . . . . . . 7 𝐵 = (Base‘𝐾)
7 dihmeetlem6.m . . . . . . 7 = (meet‘𝐾)
86, 7latmcl 18389 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → (𝑋 𝑌) ∈ 𝐵)
10 simprll 777 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑄𝐴)
11 dihmeetlem6.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
126, 11atbase 38147 . . . . . 6 (𝑄𝐴𝑄𝐵)
1310, 12syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑄𝐵)
14 simpl1r 1225 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑊𝐻)
15 dihmeetlem6.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
166, 15lhpbase 38857 . . . . . 6 (𝑊𝐻𝑊𝐵)
1714, 16syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑊𝐵)
18 dihmeetlem6.l . . . . . 6 = (le‘𝐾)
19 dihmeetlem6.j . . . . . 6 = (join‘𝐾)
206, 18, 19latjle12 18399 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵𝑄𝐵𝑊𝐵)) → (((𝑋 𝑌) 𝑊𝑄 𝑊) ↔ ((𝑋 𝑌) 𝑄) 𝑊))
213, 9, 13, 17, 20syl13anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → (((𝑋 𝑌) 𝑊𝑄 𝑊) ↔ ((𝑋 𝑌) 𝑄) 𝑊))
22 simpr 485 . . . 4 (((𝑋 𝑌) 𝑊𝑄 𝑊) → 𝑄 𝑊)
2321, 22syl6bir 253 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → (((𝑋 𝑌) 𝑄) 𝑊𝑄 𝑊))
241, 23mtod 197 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ¬ ((𝑋 𝑌) 𝑄) 𝑊)
25 simprr 771 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → 𝑄 𝑋)
266, 18, 19, 7, 11dihmeetlem5 40167 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → (𝑋 (𝑌 𝑄)) = ((𝑋 𝑌) 𝑄))
272, 4, 5, 10, 25, 26syl32anc 1378 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → (𝑋 (𝑌 𝑄)) = ((𝑋 𝑌) 𝑄))
2827breq1d 5157 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ((𝑋 (𝑌 𝑄)) 𝑊 ↔ ((𝑋 𝑌) 𝑄) 𝑊))
2924, 28mtbird 324 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ¬ (𝑋 (𝑌 𝑄)) 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  Latclat 18380  Atomscatm 38121  HLchlt 38208  LHypclh 38843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-psubsp 38362  df-pmap 38363  df-padd 38655  df-lhyp 38847
This theorem is referenced by:  dihjatc1  40170  dihmeetlem10N  40175
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