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Theorem dihmeetlem6 40837
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 778 . . 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 1221 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
32hllatd 38891 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
4 simpl2 1189 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
5 simpl3 1190 . . . . . 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 18429 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
93, 4, 5, 8syl3anc 1368 . . . . 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 777 . . . . . 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 38816 . . . . . 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 1222 . . . . . 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 39526 . . . . . 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 18439 . . . . 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 1369 . . . 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 483 . . . 4 (((𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑄 ≀ π‘Š) β†’ 𝑄 ≀ π‘Š)
2321, 22syl6bir 253 . . 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 771 . . . 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 40836 . . . 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 1375 . . 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 5153 . 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 324 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5143  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  lecple 17237  joincjn 18300  meetcmee 18301  Latclat 18420  Atomscatm 38790  HLchlt 38877  LHypclh 39512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-proset 18284  df-poset 18302  df-plt 18319  df-lub 18335  df-glb 18336  df-join 18337  df-meet 18338  df-p0 18414  df-lat 18421  df-clat 18488  df-oposet 38703  df-ol 38705  df-oml 38706  df-covers 38793  df-ats 38794  df-atl 38825  df-cvlat 38849  df-hlat 38878  df-psubsp 39031  df-pmap 39032  df-padd 39324  df-lhyp 39516
This theorem is referenced by:  dihjatc1  40839  dihmeetlem10N  40844
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