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Theorem ps-2c 36696
Description: Variation of projective geometry axiom ps-2 36646. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
2atm.l = (le‘𝐾)
2atm.j = (join‘𝐾)
2atm.m = (meet‘𝐾)
2atm.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
ps-2c (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ 𝐴)

Proof of Theorem ps-2c
StepHypRef Expression
1 simp11 1199 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝐾 ∈ HL)
2 simp12 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃𝐴)
3 simp21 1202 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑅𝐴)
41hllatd 36532 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝐾 ∈ Lat)
5 eqid 2820 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
6 2atm.a . . . . . 6 𝐴 = (Atoms‘𝐾)
75, 6atbase 36457 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
82, 7syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃 ∈ (Base‘𝐾))
9 simp13 1201 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑄𝐴)
105, 6atbase 36457 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
119, 10syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑄 ∈ (Base‘𝐾))
125, 6atbase 36457 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
133, 12syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑅 ∈ (Base‘𝐾))
14 simp31l 1292 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ¬ 𝑃 (𝑄 𝑅))
15 2atm.l . . . . 5 = (le‘𝐾)
16 2atm.j . . . . 5 = (join‘𝐾)
175, 15, 16latnlej1r 17655 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)
184, 8, 11, 13, 14, 17syl131anc 1379 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃𝑅)
19 eqid 2820 . . . 4 (LLines‘𝐾) = (LLines‘𝐾)
2016, 6, 19llni2 36680 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 𝑅) ∈ (LLines‘𝐾))
211, 2, 3, 18, 20syl31anc 1369 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑃 𝑅) ∈ (LLines‘𝐾))
22 simp22 1203 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝐴)
23 simp23 1204 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑇𝐴)
24 simp31r 1293 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝑇)
2516, 6, 19llni2 36680 . . 3 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
261, 22, 23, 24, 25syl31anc 1369 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
27 simp32 1206 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑃 𝑅) ≠ (𝑆 𝑇))
28 simp33 1207 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))
29 2atm.m . . . 4 = (meet‘𝐾)
30 eqid 2820 . . . 4 (0.‘𝐾) = (0.‘𝐾)
3115, 16, 29, 30, 6ps-2b 36650 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ (0.‘𝐾))
321, 2, 9, 3, 22, 23, 14, 24, 28, 31syl333anc 1398 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ (0.‘𝐾))
3329, 30, 6, 192llnmat 36692 . 2 (((𝐾 ∈ HL ∧ (𝑃 𝑅) ∈ (LLines‘𝐾) ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) ∧ ((𝑃 𝑅) ≠ (𝑆 𝑇) ∧ ((𝑃 𝑅) (𝑆 𝑇)) ≠ (0.‘𝐾))) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ 𝐴)
341, 21, 26, 27, 32, 33syl32anc 1374 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006   class class class wbr 5038  cfv 6327  (class class class)co 7129  Basecbs 16458  lecple 16547  joincjn 17529  meetcmee 17530  0.cp0 17622  Latclat 17630  Atomscatm 36431  HLchlt 36518  LLinesclln 36659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-proset 17513  df-poset 17531  df-plt 17543  df-lub 17559  df-glb 17560  df-join 17561  df-meet 17562  df-p0 17624  df-lat 17631  df-clat 17693  df-oposet 36344  df-ol 36346  df-oml 36347  df-covers 36434  df-ats 36435  df-atl 36466  df-cvlat 36490  df-hlat 36519  df-llines 36666
This theorem is referenced by:  cdlemg18c  37848  dia2dimlem1  38232
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