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Theorem cdleme0nex 36099
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 36020- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 35152, 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 1244 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ 𝑅 𝑊)
2 simp12 1246 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅 (𝑃 𝑄))
31, 2jca 501 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (¬ 𝑅 𝑊𝑅 (𝑃 𝑄)))
4 simp3l 1243 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅𝐴)
5 simp13 1247 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
6 ralnex 3141 . . . . . . 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 4789 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟 𝑊𝑅 𝑊))
98notbid 307 . . . . . . . . 9 (𝑟 = 𝑅 → (¬ 𝑟 𝑊 ↔ ¬ 𝑅 𝑊))
10 oveq2 6801 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
11 oveq2 6801 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1210, 11eqeq12d 2786 . . . . . . . . 9 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
139, 12anbi12d 616 . . . . . . . 8 (𝑟 = 𝑅 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1413notbid 307 . . . . . . 7 (𝑟 = 𝑅 → (¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1514rspcva 3458 . . . . . 6 ((𝑅𝐴 ∧ ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
164, 7, 15syl2anc 573 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
17 simp11 1245 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ HL)
18 hlcvl 35168 . . . . . . . 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 1248 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝐴)
21 simp22 1249 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑄𝐴)
22 simp23 1250 . . . . . . 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 35152 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1489 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2827anbi2d 614 . . . . 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 313 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
30 ianor 962 . . . . 5 (¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
31 df-3an 1073 . . . . . . . 8 ((𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)) ↔ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄)))
3231anbi2i 609 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ (¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))))
33 an12 624 . . . . . . 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 309 . . . . 5 (¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
36 pm4.62 843 . . . . 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 133 . 2 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (𝑅𝑃𝑅𝑄))
40 neanior 3035 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
4140con2bii 346 . 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  wa 382  wo 834  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  Atomscatm 35072  CvLatclc 35074  HLchlt 35159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160
This theorem is referenced by:  cdleme18c  36102  cdleme18d  36104  cdlemg17b  36471  cdlemg17h  36477
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