Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemeg46nlpq Structured version   Visualization version   GIF version

Theorem cdlemeg46nlpq 36325
 Description: Show that (𝐺‘𝑆) is not under 𝑃 ∨ 𝑄 when 𝑆 isn't. (Contributed by NM, 3-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46g.b 𝐵 = (Base‘𝐾)
cdlemef46g.l = (le‘𝐾)
cdlemef46g.j = (join‘𝐾)
cdlemef46g.m = (meet‘𝐾)
cdlemef46g.a 𝐴 = (Atoms‘𝐾)
cdlemef46g.h 𝐻 = (LHyp‘𝐾)
cdlemef46g.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef46g.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs46g.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef46g.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
cdlemef46.v 𝑉 = ((𝑄 𝑃) 𝑊)
cdlemef46.n 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
cdlemefs46.o 𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))
cdlemef46.g 𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))
Assertion
Ref Expression
cdlemeg46nlpq ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ (𝐺𝑆) (𝑃 𝑄))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑆,𝑠,𝑡,𝑥,𝑦,𝑧   𝑎,𝑏,𝑐,𝑢,𝑣,𝐴   𝐵,𝑎,𝑏,𝑐,𝑢,𝑣   𝑣,𝐷   𝐺,𝑠,𝑡,𝑥,𝑦,𝑧   𝐻,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   𝐾,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   𝑁,𝑎,𝑏,𝑐   𝑂,𝑎,𝑏,𝑐   𝑃,𝑎,𝑏,𝑐,𝑢,𝑣   𝑄,𝑎,𝑏,𝑐,𝑢,𝑣   𝑆,𝑎,𝑏,𝑐,𝑢,𝑣   𝑉,𝑎,𝑏,𝑐   𝑊,𝑎,𝑏,𝑐,𝑢,𝑣   𝑥,𝑢,𝑦,𝑧,𝑁   𝑥,𝑂,𝑦,𝑧   𝑣,𝑡   𝑢,𝑉   𝑥,𝑣,𝑦,𝑧,𝑉
Allowed substitution hints:   𝐷(𝑢,𝑡,𝑎,𝑏,𝑐)   𝑈(𝑣,𝑢,𝑎,𝑏,𝑐)   𝐸(𝑣,𝑢,𝑡,𝑠,𝑎,𝑏,𝑐)   𝐹(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠,𝑎,𝑏,𝑐)   𝐺(𝑣,𝑢,𝑎,𝑏,𝑐)   𝑁(𝑣,𝑡,𝑠)   𝑂(𝑣,𝑢,𝑡,𝑠)   𝑉(𝑡,𝑠)

Proof of Theorem cdlemeg46nlpq
StepHypRef Expression
1 simp11 1246 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp13 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simp12 1247 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp2l 1242 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑄)
54necomd 2987 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑄𝑃)
6 simp2r 1243 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
7 simp3 1133 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ 𝑆 (𝑃 𝑄))
8 simp11l 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝐾 ∈ HL)
9 simp12l 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝐴)
10 simp13l 1373 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑄𝐴)
11 cdlemef46g.j . . . . . . 7 = (join‘𝐾)
12 cdlemef46g.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1311, 12hlatjcom 35175 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
148, 9, 10, 13syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑄 𝑃))
1514breq2d 4816 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑆 (𝑃 𝑄) ↔ 𝑆 (𝑄 𝑃)))
167, 15mtbid 313 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ 𝑆 (𝑄 𝑃))
17 cdlemef46g.b . . . 4 𝐵 = (Base‘𝐾)
18 cdlemef46g.l . . . 4 = (le‘𝐾)
19 cdlemef46g.m . . . 4 = (meet‘𝐾)
20 cdlemef46g.h . . . 4 𝐻 = (LHyp‘𝐾)
21 cdlemef46.v . . . 4 𝑉 = ((𝑄 𝑃) 𝑊)
22 cdlemef46.n . . . 4 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
23 cdlemefs46.o . . . 4 𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))
24 cdlemef46.g . . . 4 𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))
2517, 18, 11, 19, 12, 20, 21, 22, 23, 24cdleme46frvlpq 36312 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝑃 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑄 𝑃)) → ¬ (𝐺𝑆) (𝑄 𝑃))
261, 2, 3, 5, 6, 16, 25syl321anc 1499 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ (𝐺𝑆) (𝑄 𝑃))
2714breq2d 4816 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ((𝐺𝑆) (𝑃 𝑄) ↔ (𝐺𝑆) (𝑄 𝑃)))
2826, 27mtbird 314 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ (𝐺𝑆) (𝑃 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ⦋csb 3674  ifcif 4230   class class class wbr 4804   ↦ cmpt 4881  ‘cfv 6049  ℩crio 6774  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  Atomscatm 35071  HLchlt 35158  LHypclh 35791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795 This theorem is referenced by:  cdlemeg46req  36337
 Copyright terms: Public domain W3C validator