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Theorem dihmeetlem3N 37096
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem3.b 𝐵 = (Base‘𝐾)
dihmeetlem3.l = (le‘𝐾)
dihmeetlem3.j = (join‘𝐾)
dihmeetlem3.m = (meet‘𝐾)
dihmeetlem3.a 𝐴 = (Atoms‘𝐾)
dihmeetlem3.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
dihmeetlem3N ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → 𝑄𝑅)

Proof of Theorem dihmeetlem3N
StepHypRef Expression
1 simp2lr 1308 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → ¬ 𝑄 𝑊)
2 oveq1 6820 . . . . . . 7 (𝑄 = 𝑅 → (𝑄 (𝑌 𝑊)) = (𝑅 (𝑌 𝑊)))
3 simpr 479 . . . . . . 7 (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) → (𝑅 (𝑌 𝑊)) = 𝑌)
42, 3sylan9eqr 2816 . . . . . 6 ((((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) ∧ 𝑄 = 𝑅) → (𝑄 (𝑌 𝑊)) = 𝑌)
5 dihmeetlem3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
6 dihmeetlem3.l . . . . . . . 8 = (le‘𝐾)
7 simp11l 1369 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ HL)
8 hllat 35153 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
97, 8syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ Lat)
10 simp2ll 1307 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄𝐴)
11 dihmeetlem3.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
125, 11atbase 35079 . . . . . . . . 9 (𝑄𝐴𝑄𝐵)
1310, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄𝐵)
14 simp12l 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑋𝐵)
15 simp12r 1372 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑌𝐵)
16 dihmeetlem3.m . . . . . . . . . 10 = (meet‘𝐾)
175, 16latmcl 17253 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
189, 14, 15, 17syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) ∈ 𝐵)
19 simp11r 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑊𝐻)
20 dihmeetlem3.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
215, 20lhpbase 35787 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
2219, 21syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑊𝐵)
235, 16latmcl 17253 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
249, 14, 22, 23syl3anc 1477 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑊) ∈ 𝐵)
25 dihmeetlem3.j . . . . . . . . . . . 12 = (join‘𝐾)
265, 6, 25latlej1 17261 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
279, 13, 24, 26syl3anc 1477 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑄 (𝑋 𝑊)))
28 simp2r 1243 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑄 (𝑋 𝑊)) = 𝑋)
2927, 28breqtrd 4830 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑋)
305, 16latmcl 17253 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
319, 15, 22, 30syl3anc 1477 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑌 𝑊) ∈ 𝐵)
325, 6, 25latlej1 17261 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑌 𝑊)))
339, 13, 31, 32syl3anc 1477 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑄 (𝑌 𝑊)))
34 simp3 1133 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑄 (𝑌 𝑊)) = 𝑌)
3533, 34breqtrd 4830 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑌)
365, 6, 16latlem12 17279 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑋𝐵𝑌𝐵)) → ((𝑄 𝑋𝑄 𝑌) ↔ 𝑄 (𝑋 𝑌)))
379, 13, 14, 15, 36syl13anc 1479 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → ((𝑄 𝑋𝑄 𝑌) ↔ 𝑄 (𝑋 𝑌)))
3829, 35, 37mpbi2and 994 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑋 𝑌))
39 simp13 1248 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) 𝑊)
405, 6, 9, 13, 18, 22, 38, 39lattrd 17259 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑊)
41403exp 1113 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) → (((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) → ((𝑄 (𝑌 𝑊)) = 𝑌𝑄 𝑊)))
424, 41syl7 74 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) → (((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) → ((((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) ∧ 𝑄 = 𝑅) → 𝑄 𝑊)))
4342exp4a 634 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) → (((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) → (𝑄 = 𝑅𝑄 𝑊))))
44433imp 1102 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → (𝑄 = 𝑅𝑄 𝑊))
4544necon3bd 2946 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → (¬ 𝑄 𝑊𝑄𝑅))
461, 45mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → 𝑄𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35053  HLchlt 35140  LHypclh 35773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-poset 17147  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-lat 17247  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-lhyp 35777
This theorem is referenced by: (None)
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