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Theorem dihmeetlem3N 37261
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 1322 . 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 6849 . . . . . . 7 (𝑄 = 𝑅 → (𝑄 (𝑌 𝑊)) = (𝑅 (𝑌 𝑊)))
3 simpr 477 . . . . . . 7 (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) → (𝑅 (𝑌 𝑊)) = 𝑌)
42, 3sylan9eqr 2821 . . . . . 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 1383 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ HL)
87hllatd 35320 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ Lat)
9 simp2ll 1321 . . . . . . . . 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 35245 . . . . . . . . 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 1385 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑋𝐵)
14 simp12r 1386 . . . . . . . . 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 17318 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
178, 13, 14, 16syl3anc 1490 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) ∈ 𝐵)
18 simp11r 1384 . . . . . . . . 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 35954 . . . . . . . . 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 17318 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
238, 13, 21, 22syl3anc 1490 . . . . . . . . . . 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 17326 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
268, 12, 23, 25syl3anc 1490 . . . . . . . . . 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 1257 . . . . . . . . . 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 4835 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑋)
295, 15latmcl 17318 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
308, 14, 21, 29syl3anc 1490 . . . . . . . . . . 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 17326 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑌 𝑊)))
328, 12, 30, 31syl3anc 1490 . . . . . . . . . 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 1168 . . . . . . . . . 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 4835 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑌)
355, 6, 15latlem12 17344 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑋𝐵𝑌𝐵)) → ((𝑄 𝑋𝑄 𝑌) ↔ 𝑄 (𝑋 𝑌)))
368, 12, 13, 14, 35syl13anc 1491 . . . . . . . . 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 703 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑋 𝑌))
38 simp13 1262 . . . . . . . 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 17324 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑊)
40393exp 1148 . . . . . 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 422 . . . 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 1137 . . 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 2951 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  joincjn 17210  meetcmee 17211  Latclat 17311  Atomscatm 35219  HLchlt 35306  LHypclh 35940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-poset 17212  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-lat 17312  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307  df-lhyp 35944
This theorem is referenced by: (None)
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