Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme36a Structured version   Visualization version   GIF version

Theorem cdleme36a 36439
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013.)
Hypotheses
Ref Expression
cdleme36.b 𝐵 = (Base‘𝐾)
cdleme36.l = (le‘𝐾)
cdleme36.j = (join‘𝐾)
cdleme36.m = (meet‘𝐾)
cdleme36.a 𝐴 = (Atoms‘𝐾)
cdleme36.h 𝐻 = (LHyp‘𝐾)
cdleme36.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme36.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
Assertion
Ref Expression
cdleme36a ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → ¬ 𝑅 (𝑡 𝐸))

Proof of Theorem cdleme36a
StepHypRef Expression
1 simp3r 1259 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → ¬ 𝑡 (𝑃 𝑄))
2 simp11l 1383 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝐾 ∈ HL)
3 simp22l 1391 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑅𝐴)
4 simp3ll 1325 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑡𝐴)
5 simp11 1260 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 simp12 1261 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
7 simp13 1262 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑄𝐴)
8 simp21 1263 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑃𝑄)
9 cdleme36.l . . . . . 6 = (le‘𝐾)
10 cdleme36.j . . . . . 6 = (join‘𝐾)
11 cdleme36.m . . . . . 6 = (meet‘𝐾)
12 cdleme36.a . . . . . 6 𝐴 = (Atoms‘𝐾)
13 cdleme36.h . . . . . 6 𝐻 = (LHyp‘𝐾)
14 cdleme36.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
159, 10, 11, 12, 13, 14cdleme0a 36190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
165, 6, 7, 8, 15syl112anc 1493 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑈𝐴)
17 simp12l 1385 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑃𝐴)
18 simp22 1264 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
199, 10, 11, 12, 13, 14cdleme0c 36192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑈𝑅)
205, 17, 7, 18, 19syl121anc 1494 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑈𝑅)
2120necomd 2992 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑅𝑈)
229, 10, 12hlatexch2 35375 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑡𝐴𝑈𝐴) ∧ 𝑅𝑈) → (𝑅 (𝑡 𝑈) → 𝑡 (𝑅 𝑈)))
232, 3, 4, 16, 21, 22syl131anc 1502 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑅 (𝑡 𝑈) → 𝑡 (𝑅 𝑈)))
24 simp3l 1258 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
25 cdleme36.e . . . . . 6 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
269, 10, 11, 12, 13, 14, 25cdleme1 36206 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊))) → (𝑡 𝐸) = (𝑡 𝑈))
275, 17, 7, 24, 26syl13anc 1491 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑡 𝐸) = (𝑡 𝑈))
2827breq2d 4823 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑅 (𝑡 𝐸) ↔ 𝑅 (𝑡 𝑈)))
29 simp23 1265 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
309, 10, 11, 12, 13, 14cdleme4 36217 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑅 𝑈))
315, 17, 7, 18, 29, 30syl131anc 1502 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑅 𝑈))
3231breq2d 4823 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑡 (𝑃 𝑄) ↔ 𝑡 (𝑅 𝑈)))
3323, 28, 323imtr4d 285 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑅 (𝑡 𝐸) → 𝑡 (𝑃 𝑄)))
341, 33mtod 189 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → ¬ 𝑅 (𝑡 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4811  cfv 6070  (class class class)co 6846  Basecbs 16146  lecple 16237  joincjn 17226  meetcmee 17227  Atomscatm 35242  HLchlt 35329  LHypclh 35963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-proset 17210  df-poset 17228  df-plt 17240  df-lub 17256  df-glb 17257  df-join 17258  df-meet 17259  df-p0 17321  df-p1 17322  df-lat 17328  df-clat 17390  df-oposet 35155  df-ol 35157  df-oml 35158  df-covers 35245  df-ats 35246  df-atl 35277  df-cvlat 35301  df-hlat 35330  df-psubsp 35482  df-pmap 35483  df-padd 35775  df-lhyp 35967
This theorem is referenced by:  cdleme36m  36440
  Copyright terms: Public domain W3C validator