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Theorem cdleme20l1 39849
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, π‘Œ, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l ≀ = (leβ€˜πΎ)
cdleme19.j ∨ = (joinβ€˜πΎ)
cdleme19.m ∧ = (meetβ€˜πΎ)
cdleme19.a 𝐴 = (Atomsβ€˜πΎ)
cdleme19.h 𝐻 = (LHypβ€˜πΎ)
cdleme19.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme19.f 𝐹 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme19.g 𝐺 = ((𝑇 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ π‘Š)))
cdleme19.d 𝐷 = ((𝑅 ∨ 𝑆) ∧ π‘Š)
cdleme19.y π‘Œ = ((𝑅 ∨ 𝑇) ∧ π‘Š)
cdleme20.v 𝑉 = ((𝑆 ∨ 𝑇) ∧ π‘Š)
Assertion
Ref Expression
cdleme20l1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝐹 ∨ 𝐷) ∈ (LLinesβ€˜πΎ))

Proof of Theorem cdleme20l1
StepHypRef Expression
1 simp11l 1281 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ HL)
2 simp11 1200 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3 simp12 1201 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
4 simp13 1202 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
5 simp22 1204 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑆 ∈ 𝐴)
6 simp23 1205 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑆 ≀ π‘Š)
75, 6jca 510 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
8 simp31 1206 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 β‰  𝑄)
9 simp32 1207 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
10 cdleme19.l . . . 4 ≀ = (leβ€˜πΎ)
11 cdleme19.j . . . 4 ∨ = (joinβ€˜πΎ)
12 cdleme19.m . . . 4 ∧ = (meetβ€˜πΎ)
13 cdleme19.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
14 cdleme19.h . . . 4 𝐻 = (LHypβ€˜πΎ)
15 cdleme19.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
16 cdleme19.f . . . 4 𝐹 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
1710, 11, 12, 13, 14, 15, 16cdleme3fa 39765 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 ∈ 𝐴)
182, 3, 4, 7, 8, 9, 17syl132anc 1385 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 ∈ 𝐴)
19 simp11r 1282 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ π‘Š ∈ 𝐻)
20 simp21 1203 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ∈ 𝐴)
21 simp33 1208 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
22 cdleme19.d . . . 4 𝐷 = ((𝑅 ∨ 𝑆) ∧ π‘Š)
2310, 11, 12, 13, 14, 22cdlemeda 39827 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐷 ∈ 𝐴)
241, 19, 5, 6, 20, 21, 9, 23syl223anc 1393 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐷 ∈ 𝐴)
2510, 11, 12, 13, 14, 15, 16, 16, 22, 22cdleme19c 39834 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 β‰  𝐷)
261, 19, 3, 4, 7, 20, 8, 9, 25syl233anc 1396 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 β‰  𝐷)
27 eqid 2725 . . 3 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
2811, 13, 27llni2 39041 . 2 (((𝐾 ∈ HL ∧ 𝐹 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐹 β‰  𝐷) β†’ (𝐹 ∨ 𝐷) ∈ (LLinesβ€˜πΎ))
291, 18, 24, 26, 28syl31anc 1370 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝐹 ∨ 𝐷) ∈ (LLinesβ€˜πΎ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  lecple 17239  joincjn 18302  meetcmee 18303  Atomscatm 38791  HLchlt 38878  LLinesclln 39020  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-llines 39027  df-lines 39030  df-psubsp 39032  df-pmap 39033  df-padd 39325  df-lhyp 39517
This theorem is referenced by:  cdleme20l2  39850  cdleme20l  39851
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