Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihmeetlem6 Structured version   Visualization version   GIF version

Theorem dihmeetlem6 39822
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihmeetlem6.b 𝐡 = (Baseβ€˜πΎ)
dihmeetlem6.l ≀ = (leβ€˜πΎ)
dihmeetlem6.h 𝐻 = (LHypβ€˜πΎ)
dihmeetlem6.j ∨ = (joinβ€˜πΎ)
dihmeetlem6.m ∧ = (meetβ€˜πΎ)
dihmeetlem6.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
dihmeetlem6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ Β¬ (𝑋 ∧ (π‘Œ ∨ 𝑄)) ≀ π‘Š)

Proof of Theorem dihmeetlem6
StepHypRef Expression
1 simprlr 779 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑄 ≀ π‘Š)
2 simpl1l 1225 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
32hllatd 37876 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
4 simpl2 1193 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
5 simpl3 1194 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ π‘Œ ∈ 𝐡)
6 dihmeetlem6.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
7 dihmeetlem6.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
86, 7latmcl 18337 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
93, 4, 5, 8syl3anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
10 simprll 778 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
11 dihmeetlem6.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
126, 11atbase 37801 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1310, 12syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
14 simpl1r 1226 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ π‘Š ∈ 𝐻)
15 dihmeetlem6.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
166, 15lhpbase 38511 . . . . . 6 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
1714, 16syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ π‘Š ∈ 𝐡)
18 dihmeetlem6.l . . . . . 6 ≀ = (leβ€˜πΎ)
19 dihmeetlem6.j . . . . . 6 ∨ = (joinβ€˜πΎ)
206, 18, 19latjle12 18347 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑋 ∧ π‘Œ) ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ (((𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑄 ≀ π‘Š) ↔ ((𝑋 ∧ π‘Œ) ∨ 𝑄) ≀ π‘Š))
213, 9, 13, 17, 20syl13anc 1373 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ (((𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑄 ≀ π‘Š) ↔ ((𝑋 ∧ π‘Œ) ∨ 𝑄) ≀ π‘Š))
22 simpr 486 . . . 4 (((𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑄 ≀ π‘Š) β†’ 𝑄 ≀ π‘Š)
2321, 22syl6bir 254 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ (((𝑋 ∧ π‘Œ) ∨ 𝑄) ≀ π‘Š β†’ 𝑄 ≀ π‘Š))
241, 23mtod 197 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ Β¬ ((𝑋 ∧ π‘Œ) ∨ 𝑄) ≀ π‘Š)
25 simprr 772 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ≀ 𝑋)
266, 18, 19, 7, 11dihmeetlem5 39821 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑄)) = ((𝑋 ∧ π‘Œ) ∨ 𝑄))
272, 4, 5, 10, 25, 26syl32anc 1379 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑄)) = ((𝑋 ∧ π‘Œ) ∨ 𝑄))
2827breq1d 5119 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ ((𝑋 ∧ (π‘Œ ∨ 𝑄)) ≀ π‘Š ↔ ((𝑋 ∧ π‘Œ) ∨ 𝑄) ≀ π‘Š))
2924, 28mtbird 325 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ Β¬ (𝑋 ∧ (π‘Œ ∨ 𝑄)) ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  lecple 17148  joincjn 18208  meetcmee 18209  Latclat 18328  Atomscatm 37775  HLchlt 37862  LHypclh 38497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-lat 18329  df-clat 18396  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501
This theorem is referenced by:  dihjatc1  39824  dihmeetlem10N  39829
  Copyright terms: Public domain W3C validator