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Theorem cdleme17b 39816
Description: Lemma leading to cdleme17c 39817. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l ≀ = (leβ€˜πΎ)
cdleme17.j ∨ = (joinβ€˜πΎ)
cdleme17.m ∧ = (meetβ€˜πΎ)
cdleme17.a 𝐴 = (Atomsβ€˜πΎ)
cdleme17.h 𝐻 = (LHypβ€˜πΎ)
cdleme17.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme17.f 𝐹 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme17.g 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme17.c 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
Assertion
Ref Expression
cdleme17b (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))

Proof of Theorem cdleme17b
StepHypRef Expression
1 simp33 1208 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
2 eqid 2725 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 cdleme17.l . . 3 ≀ = (leβ€˜πΎ)
4 simpl1l 1221 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐾 ∈ HL)
54hllatd 38892 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐾 ∈ Lat)
6 simpl32 1252 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ∈ 𝐴)
7 cdleme17.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
82, 7atbase 38817 . . . 4 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
96, 8syl 17 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
10 simpl2l 1223 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ∈ 𝐴)
11 cdleme17.j . . . . 5 ∨ = (joinβ€˜πΎ)
122, 11, 7hlatjcl 38895 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
134, 10, 6, 12syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
14 simpl31 1251 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑄 ∈ 𝐴)
152, 11, 7hlatjcl 38895 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
164, 10, 14, 15syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
173, 11, 7hlatlej2 38904 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ 𝑆 ≀ (𝑃 ∨ 𝑆))
184, 10, 6, 17syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑆))
19 simpl1r 1222 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ π‘Š ∈ 𝐻)
20 simpl2r 1224 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ Β¬ 𝑃 ≀ π‘Š)
21 cdleme17.m . . . . . 6 ∧ = (meetβ€˜πΎ)
22 cdleme17.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
23 cdleme17.c . . . . . 6 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
243, 11, 21, 7, 22, 23cdleme8 39779 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
254, 19, 10, 20, 6, 24syl221anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
263, 11, 7hlatlej1 38903 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
274, 10, 14, 26syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
28 simpr 483 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐢 ≀ (𝑃 ∨ 𝑄))
292, 7atbase 38817 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
3010, 29syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
312, 11, 21, 7, 22, 23cdleme9b 39781 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ π‘Š ∈ 𝐻)) β†’ 𝐢 ∈ (Baseβ€˜πΎ))
324, 10, 6, 19, 31syl13anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐢 ∈ (Baseβ€˜πΎ))
332, 3, 11latjle12 18441 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝐢 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄)))
345, 30, 32, 16, 33syl13anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄)))
3527, 28, 34mpbi2and 710 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄))
3625, 35eqbrtrrd 5167 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑆) ≀ (𝑃 ∨ 𝑄))
372, 3, 5, 9, 13, 16, 18, 36lattrd 18437 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄))
381, 37mtand 814 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))
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 6543  (class class class)co 7416  Basecbs 17179  lecple 17239  joincjn 18302  meetcmee 18303  Latclat 18422  Atomscatm 38791  HLchlt 38878  LHypclh 39513
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 7738
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 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-clat 18490  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-psubsp 39032  df-pmap 39033  df-padd 39325  df-lhyp 39517
This theorem is referenced by:  cdleme17c  39817
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