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Theorem dihmeetlem3N 39768
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 1241 . 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 7364 . . . . . . 7 (𝑄 = 𝑅 → (𝑄 (𝑌 𝑊)) = (𝑅 (𝑌 𝑊)))
3 simpr 485 . . . . . . 7 (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌) → (𝑅 (𝑌 𝑊)) = 𝑌)
42, 3sylan9eqr 2798 . . . . . 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 1284 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ HL)
87hllatd 37826 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝐾 ∈ Lat)
9 simp2ll 1240 . . . . . . . . 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 37751 . . . . . . . . 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 1286 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑋𝐵)
14 simp12r 1287 . . . . . . . . 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 18329 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
178, 13, 14, 16syl3anc 1371 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → (𝑋 𝑌) ∈ 𝐵)
18 simp11r 1285 . . . . . . . . 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 38461 . . . . . . . . 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 18329 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
238, 13, 21, 22syl3anc 1371 . . . . . . . . . . 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 18337 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
268, 12, 23, 25syl3anc 1371 . . . . . . . . . 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 1200 . . . . . . . . . 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 5131 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑋)
295, 15latmcl 18329 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
308, 14, 21, 29syl3anc 1371 . . . . . . . . . . 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 18337 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑌 𝑊)))
328, 12, 30, 31syl3anc 1371 . . . . . . . . . 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 1138 . . . . . . . . . 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 5131 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑌)
355, 6, 15latlem12 18355 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑋𝐵𝑌𝐵)) → ((𝑄 𝑋𝑄 𝑌) ↔ 𝑄 (𝑋 𝑌)))
368, 12, 13, 14, 35syl13anc 1372 . . . . . . . . 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 710 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 (𝑋 𝑌))
38 simp13 1205 . . . . . . . 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 18335 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ (𝑄 (𝑌 𝑊)) = 𝑌) → 𝑄 𝑊)
40393exp 1119 . . . . . 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 432 . . . 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 1111 . . 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 2957 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  HLchlt 37812  LHypclh 38447
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-lat 18321  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lhyp 38451
This theorem is referenced by: (None)
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