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Theorem 4atexlem7 35883
Description: Whenever there are at least 4 atoms under 𝑃 𝑄 (specifically, 𝑃, 𝑄, 𝑟, and (𝑃 𝑄) 𝑊), there are also at least 4 atoms under 𝑃 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 35152, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). With a longer proof, the condition ¬ 𝑆 (𝑃 𝑄) could be eliminated (see 4atex 35884), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4that.l = (le‘𝐾)
4that.j = (join‘𝐾)
4that.a 𝐴 = (Atoms‘𝐾)
4that.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
4atexlem7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Distinct variable groups:   𝑧,𝑟,𝐴   𝐻,𝑟   ,𝑟,𝑧   𝐾,𝑟,𝑧   ,𝑟,𝑧   𝑃,𝑟,𝑧   𝑄,𝑟,𝑧   𝑆,𝑟,𝑧   𝑊,𝑟,𝑧
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atexlem7
StepHypRef Expression
1 simp11l 1368 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp1r1 1353 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
323ad2ant1 1127 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp1r2 1354 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
543ad2ant1 1127 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
6 simp2 1131 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → 𝑟𝐴)
7 simp3l 1243 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ 𝑟 𝑊)
86, 7jca 501 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → (𝑟𝐴 ∧ ¬ 𝑟 𝑊))
9 simp1r3 1355 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝐴)
1093ad2ant1 1127 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → 𝑆𝐴)
11 simp3r 1244 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → (𝑃 𝑟) = (𝑄 𝑟))
12 simp12 1246 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → 𝑃𝑄)
13 simp13 1247 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ 𝑆 (𝑃 𝑄))
14 4that.l . . . . . . 7 = (le‘𝐾)
15 4that.j . . . . . . 7 = (join‘𝐾)
16 eqid 2771 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
17 4that.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
18 4that.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
1914, 15, 16, 17, 184atexlemex6 35882 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑟) = (𝑄 𝑟) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
201, 3, 5, 8, 10, 11, 12, 13, 19syl323anc 1506 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
2120rexlimdv3a 3181 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
22213exp 1112 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) → (𝑃𝑄 → (¬ 𝑆 (𝑃 𝑄) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))))
23223impd 1441 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
24233impia 1109 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35072  HLchlt 35159  LHypclh 35792
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-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lhyp 35796
This theorem is referenced by:  4atex  35884  cdleme21i  36144
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