Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme20k Structured version   Visualization version   GIF version

Theorem cdleme20k 37335
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 20-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l = (le‘𝐾)
cdleme19.j = (join‘𝐾)
cdleme19.m = (meet‘𝐾)
cdleme19.a 𝐴 = (Atoms‘𝐾)
cdleme19.h 𝐻 = (LHyp‘𝐾)
cdleme19.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme19.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme19.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme19.d 𝐷 = ((𝑅 𝑆) 𝑊)
cdleme19.y 𝑌 = ((𝑅 𝑇) 𝑊)
cdleme20.v 𝑉 = ((𝑆 𝑇) 𝑊)
Assertion
Ref Expression
cdleme20k ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝐷) ≠ (𝑃 𝑄))

Proof of Theorem cdleme20k
StepHypRef Expression
1 simp11 1195 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp12 1196 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑃𝐴)
3 simp13 1197 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑄𝐴)
4 simp2r 1192 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
5 simp2l 1191 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
6 simp3r 1194 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
7 simp3l 1193 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
8 cdleme19.l . . . 4 = (le‘𝐾)
9 cdleme19.j . . . 4 = (join‘𝐾)
10 cdleme19.m . . . 4 = (meet‘𝐾)
11 cdleme19.a . . . 4 𝐴 = (Atoms‘𝐾)
12 cdleme19.h . . . 4 𝐻 = (LHyp‘𝐾)
13 cdleme19.d . . . 4 𝐷 = ((𝑅 𝑆) 𝑊)
148, 9, 10, 11, 12, 13cdlemednpq 37315 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐷 (𝑃 𝑄))
151, 2, 3, 4, 5, 6, 7, 14syl133anc 1385 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝐷 (𝑃 𝑄))
16 simp11l 1276 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
1716hllatd 36380 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
18 simp11r 1277 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
19 simp2ll 1232 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑆𝐴)
20 cdleme19.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
21 cdleme19.f . . . . . . 7 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
22 eqid 2818 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
238, 9, 10, 11, 12, 20, 21, 22cdleme1b 37242 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → 𝐹 ∈ (Base‘𝐾))
2416, 18, 2, 3, 19, 23syl23anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐹 ∈ (Base‘𝐾))
25 simp2rl 1234 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
268, 9, 10, 11, 12, 13, 22cdlemedb 37313 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴)) → 𝐷 ∈ (Base‘𝐾))
2716, 18, 25, 19, 26syl22anc 834 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐷 ∈ (Base‘𝐾))
2822, 8, 9latlej2 17659 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ 𝐷 ∈ (Base‘𝐾)) → 𝐷 (𝐹 𝐷))
2917, 24, 27, 28syl3anc 1363 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐷 (𝐹 𝐷))
30 breq2 5061 . . . 4 ((𝐹 𝐷) = (𝑃 𝑄) → (𝐷 (𝐹 𝐷) ↔ 𝐷 (𝑃 𝑄)))
3129, 30syl5ibcom 246 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝐹 𝐷) = (𝑃 𝑄) → 𝐷 (𝑃 𝑄)))
3231necon3bd 3027 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (¬ 𝐷 (𝑃 𝑄) → (𝐹 𝐷) ≠ (𝑃 𝑄)))
3315, 32mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝐷) ≠ (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366  LHypclh 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004
This theorem is referenced by:  cdleme20l  37338
  Copyright terms: Public domain W3C validator