Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemc Structured version   Visualization version   GIF version
Theorem cdlemc 39068
Description: Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l ≀ = (leβ€˜πΎ)
cdlemc3.j ∨ = (joinβ€˜πΎ)
cdlemc3.m ∧ = (meetβ€˜πΎ)
cdlemc3.a 𝐴 = (Atomsβ€˜πΎ)
cdlemc3.h 𝐻 = (LHypβ€˜πΎ)
cdlemc3.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemc3.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemc (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
Proof of Theorem cdlemc
StepHypRef Expression
1 simpl1 1192 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simpl2 1193 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)))
3 simpr 486 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘ƒ) = 𝑃)
4 cdlemc3.l . . . 4 ≀ = (leβ€˜πΎ)
5 cdlemc3.j . . . 4 ∨ = (joinβ€˜πΎ)
6 cdlemc3.m . . . 4 ∧ = (meetβ€˜πΎ)
7 cdlemc3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
8 cdlemc3.h . . . 4 𝐻 = (LHypβ€˜πΎ)
9 cdlemc3.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 cdlemc3.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
114, 5, 6, 7, 8, 9, 10cdlemc6 39067 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
121, 2, 3, 11syl3anc 1372 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
13 simpl1 1192 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
14 simpl2 1193 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)))
15 simpl3 1194 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ)))
16 simpr 486 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ (πΉβ€˜π‘ƒ) β‰  𝑃)
174, 5, 6, 7, 8, 9, 10cdlemc5 39066 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ)) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
1813, 14, 15, 16, 17syl112anc 1375 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
1912, 18pm2.61dane 3030 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  meetcmee 18265  Atomscatm 38133  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  trLctrl 39029
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030
This theorem is referenced by:  cdlemd6  39074  cdlemg4e  39485  cdlemg43  39601
  Copyright terms: Public domain W3C validator