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Theorem cdleme0nex 36098
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 36019- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 35151, 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 1245 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ 𝑅 𝑊)
2 simp12 1247 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅 (𝑃 𝑄))
31, 2jca 555 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (¬ 𝑅 𝑊𝑅 (𝑃 𝑄)))
4 simp3l 1244 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅𝐴)
5 simp13 1248 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
6 ralnex 3130 . . . . . . 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 4807 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟 𝑊𝑅 𝑊))
98notbid 307 . . . . . . . . 9 (𝑟 = 𝑅 → (¬ 𝑟 𝑊 ↔ ¬ 𝑅 𝑊))
10 oveq2 6822 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
11 oveq2 6822 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1210, 11eqeq12d 2775 . . . . . . . . 9 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
139, 12anbi12d 749 . . . . . . . 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 3447 . . . . . 6 ((𝑅𝐴 ∧ ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
164, 7, 15syl2anc 696 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
17 simp11 1246 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ HL)
18 hlcvl 35167 . . . . . . . 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 1249 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝐴)
21 simp22 1250 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑄𝐴)
22 simp23 1251 . . . . . . 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 35151 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2827anbi2d 742 . . . . 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 510 . . . . 5 (¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
31 df-3an 1074 . . . . . . . 8 ((𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)) ↔ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄)))
3231anbi2i 732 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ (¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))))
33 an12 873 . . . . . . 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 pm4.62 434 . . . . 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 3024 . . 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  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6814  lecple 16170  joincjn 17165  Atomscatm 35071  CvLatclc 35073  HLchlt 35158
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-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-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-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159
This theorem is referenced by:  cdleme18c  36101  cdleme18d  36103  cdlemg17b  36470  cdlemg17h  36476
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