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Theorem dihmeetlem3N 38594
 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 1238 . 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 7146 . . . . . . 7 (𝑄 = 𝑅 → (𝑄 (𝑌 𝑊)) = (𝑅 (𝑌 𝑊)))
3 simpr 488 . . . . . . 7 (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) → (𝑅 (𝑌 𝑊)) = 𝑌)
42, 3sylan9eqr 2858 . . . . . 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 1281 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ HL)
87hllatd 36653 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ Lat)
9 simp2ll 1237 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄𝐴)
10 dihmeetlem3.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
115, 10atbase 36578 . . . . . . . . 9 (𝑄𝐴𝑄𝐵)
129, 11syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄𝐵)
13 simp12l 1283 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑋𝐵)
14 simp12r 1284 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑌𝐵)
15 dihmeetlem3.m . . . . . . . . . 10 = (meet‘𝐾)
165, 15latmcl 17657 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
178, 13, 14, 16syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) ∈ 𝐵)
18 simp11r 1282 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑊𝐻)
19 dihmeetlem3.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
205, 19lhpbase 37287 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
2118, 20syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑊𝐵)
225, 15latmcl 17657 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
238, 13, 21, 22syl3anc 1368 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑊) ∈ 𝐵)
24 dihmeetlem3.j . . . . . . . . . . . 12 = (join‘𝐾)
255, 6, 24latlej1 17665 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
268, 12, 23, 25syl3anc 1368 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑄 (𝑋 𝑊)))
27 simp2r 1197 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑄 (𝑋 𝑊)) = 𝑋)
2826, 27breqtrd 5059 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑋)
295, 15latmcl 17657 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
308, 14, 21, 29syl3anc 1368 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑌 𝑊) ∈ 𝐵)
315, 6, 24latlej1 17665 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑌 𝑊)))
328, 12, 30, 31syl3anc 1368 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑄 (𝑌 𝑊)))
33 simp3 1135 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑄 (𝑌 𝑊)) = 𝑌)
3432, 33breqtrd 5059 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑌)
355, 6, 15latlem12 17683 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑋𝐵𝑌𝐵)) → ((𝑄 𝑋𝑄 𝑌) ↔ 𝑄 (𝑋 𝑌)))
368, 12, 13, 14, 35syl13anc 1369 . . . . . . . . 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)))
3728, 34, 36mpbi2and 711 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑋 𝑌))
38 simp13 1202 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) 𝑊)
395, 6, 8, 12, 17, 21, 37, 38lattrd 17663 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑊)
40393exp 1116 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) → (((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) → ((𝑄 (𝑌 𝑊)) = 𝑌𝑄 𝑊)))
414, 40syl7 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)))
4241exp4a 435 . . . 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))))
43423imp 1108 . . 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))
4443necon3bd 3004 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → (¬ 𝑄 𝑊𝑄𝑅))
451, 44mpd 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  Latclat 17650  Atomscatm 36552  HLchlt 36639  LHypclh 37273 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-lat 17651  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640  df-lhyp 37277 This theorem is referenced by: (None)
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