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Theorem cdleme20l1 38829
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 1285 . 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 1204 . . 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 1205 . . 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 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†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
5 simp22 1208 . . . 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 1209 . . . 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 513 . . 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 1210 . . 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 1211 . . 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 38745 . . 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 1389 . 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 1286 . . 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 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†’ 𝑅 ∈ 𝐴)
21 simp33 1212 . . 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 38807 . . 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 1397 . 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 38814 . . 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 1400 . 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 2733 . . 3 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
2811, 13, 27llni2 38021 . 2 (((𝐾 ∈ HL ∧ 𝐹 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐹 β‰  𝐷) β†’ (𝐹 ∨ 𝐷) ∈ (LLinesβ€˜πΎ))
291, 18, 24, 26, 28syl31anc 1374 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  lecple 17145  joincjn 18205  meetcmee 18206  Atomscatm 37771  HLchlt 37858  LLinesclln 38000  LHypclh 38493
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859  df-llines 38007  df-lines 38010  df-psubsp 38012  df-pmap 38013  df-padd 38305  df-lhyp 38497
This theorem is referenced by:  cdleme20l2  38830  cdleme20l  38831
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