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Theorem 4atexlem7 34838
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 34107, 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 34839), 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 1170 . . . . . 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 1155 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
323ad2ant1 1080 . . . . . 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 1156 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
543ad2ant1 1080 . . . . . 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 1060 . . . . . . 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 1087 . . . . . . 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 554 . . . . . 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 1157 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴)) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝐴)
1093ad2ant1 1080 . . . . . 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 1088 . . . . . 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 1090 . . . . . 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 1091 . . . . . 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 2621 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
17 4that.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
18 4that.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
1914, 15, 16, 17, 184atexlemex6 34837 . . . . . 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 1353 . . . . 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 3026 . . . 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 1261 . . 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 1278 . 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 1258 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  lecple 15869  joincjn 16865  meetcmee 16866  Atomscatm 34027  HLchlt 34114  LHypclh 34747
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-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262  df-lhyp 34751
This theorem is referenced by:  4atex  34839  cdleme21i  35100
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