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Theorem cdleme0nex 35057
Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 34978- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 34110, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme0nex.l = (le‘𝐾)
cdleme0nex.j = (join‘𝐾)
cdleme0nex.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdleme0nex (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
Distinct variable groups:   𝐴,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑊,𝑟
Allowed substitution hint:   𝐾(𝑟)

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 1088 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ 𝑅 𝑊)
2 simp12 1090 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅 (𝑃 𝑄))
31, 2jca 554 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (¬ 𝑅 𝑊𝑅 (𝑃 𝑄)))
4 simp3l 1087 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅𝐴)
5 simp13 1091 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
6 ralnex 2986 . . . . . . 7 (∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
75, 6sylibr 224 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
8 breq1 4616 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟 𝑊𝑅 𝑊))
98notbid 308 . . . . . . . . 9 (𝑟 = 𝑅 → (¬ 𝑟 𝑊 ↔ ¬ 𝑅 𝑊))
10 oveq2 6612 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
11 oveq2 6612 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1210, 11eqeq12d 2636 . . . . . . . . 9 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
139, 12anbi12d 746 . . . . . . . 8 (𝑟 = 𝑅 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1413notbid 308 . . . . . . 7 (𝑟 = 𝑅 → (¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1514rspcva 3293 . . . . . 6 ((𝑅𝐴 ∧ ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
164, 7, 15syl2anc 692 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
17 simp11 1089 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ HL)
18 hlcvl 34126 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ CvLat)
20 simp21 1092 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝐴)
21 simp22 1093 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑄𝐴)
22 simp23 1094 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝑄)
23 cdleme0nex.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
24 cdleme0nex.l . . . . . . . 8 = (le‘𝐾)
25 cdleme0nex.j . . . . . . . 8 = (join‘𝐾)
2623, 24, 25cvlsupr2 34110 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1336 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2827anbi2d 739 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ↔ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
2916, 28mtbid 314 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
30 ianor 509 . . . . 5 (¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
31 df-3an 1038 . . . . . . . 8 ((𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)) ↔ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄)))
3231anbi2i 729 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ (¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))))
33 an12 837 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3432, 33bitri 264 . . . . . 6 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3534notbii 310 . . . . 5 (¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
36 pm4.62 435 . . . . 5 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3730, 35, 363bitr4ri 293 . . . 4 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
3829, 37sylibr 224 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
393, 38mt2d 131 . 2 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (𝑅𝑃𝑅𝑄))
40 neanior 2882 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
4140con2bii 347 . 2 ((𝑅 = 𝑃𝑅 = 𝑄) ↔ ¬ (𝑅𝑃𝑅𝑄))
4239, 41sylibr 224 1 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  lecple 15869  joincjn 16865  Atomscatm 34030  CvLatclc 34032  HLchlt 34117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118
This theorem is referenced by:  cdleme18c  35060  cdleme18d  35062  cdlemg17b  35430  cdlemg17h  35436
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