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Theorem dihmeetlem6 40693
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 777 . . 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 38747 . . . . 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 18405 . . . . . 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 776 . . . . . 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 38672 . . . . . 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 39382 . . . . . 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 18415 . . . . 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 484 . . . 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 770 . . . 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 40692 . . . 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 5151 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  Latclat 18396  Atomscatm 38646  HLchlt 38733  LHypclh 39368
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372
This theorem is referenced by:  dihjatc1  40695  dihmeetlem10N  40700
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