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Theorem ps-2b 35289
Description: Variation of projective geometry axiom ps-2 35285. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
ps-2b.l = (le‘𝐾)
ps-2b.j = (join‘𝐾)
ps-2b.m = (meet‘𝐾)
ps-2b.z 0 = (0.‘𝐾)
ps-2b.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
ps-2b (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )

Proof of Theorem ps-2b
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 simp11 1245 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝐾 ∈ HL)
2 simp12 1246 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃𝐴)
3 simp13 1247 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑄𝐴)
4 simp21 1248 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑅𝐴)
52, 3, 43jca 1122 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
6 simp22 1249 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝐴)
7 simp23 1250 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑇𝐴)
86, 7jca 501 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆𝐴𝑇𝐴))
9 simp31 1251 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ¬ 𝑃 (𝑄 𝑅))
10 simp32 1252 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝑇)
119, 10jca 501 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇))
12 simp33 1253 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))
13 ps-2b.l . . . 4 = (le‘𝐾)
14 ps-2b.j . . . 4 = (join‘𝐾)
15 ps-2b.a . . . 4 𝐴 = (Atoms‘𝐾)
1613, 14, 15ps-2 35285 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
171, 5, 8, 11, 12, 16syl32anc 1484 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
18 simp111 1386 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝐾 ∈ HL)
19 hlatl 35167 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2018, 19syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝐾 ∈ AtLat)
2118hllatd 35171 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝐾 ∈ Lat)
22 simp112 1387 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑃𝐴)
23 simp121 1389 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑅𝐴)
24 eqid 2771 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2524, 14, 15hlatjcl 35174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
2618, 22, 23, 25syl3anc 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → (𝑃 𝑅) ∈ (Base‘𝐾))
27 simp122 1390 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑆𝐴)
28 simp123 1391 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑇𝐴)
2924, 14, 15hlatjcl 35174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
3018, 27, 28, 29syl3anc 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → (𝑆 𝑇) ∈ (Base‘𝐾))
31 ps-2b.m . . . . . 6 = (meet‘𝐾)
3224, 31latmcl 17260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾))
3321, 26, 30, 32syl3anc 1476 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾))
34 simp2 1131 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑢𝐴)
35 simp3 1132 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
3624, 15atbase 35096 . . . . . . 7 (𝑢𝐴𝑢 ∈ (Base‘𝐾))
3734, 36syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑢 ∈ (Base‘𝐾))
3824, 13, 31latlem12 17286 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑢 ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)) ↔ 𝑢 ((𝑃 𝑅) (𝑆 𝑇))))
3921, 37, 26, 30, 38syl13anc 1478 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → ((𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)) ↔ 𝑢 ((𝑃 𝑅) (𝑆 𝑇))))
4035, 39mpbid 222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑢 ((𝑃 𝑅) (𝑆 𝑇)))
41 ps-2b.z . . . . 5 0 = (0.‘𝐾)
4224, 13, 41, 15atlen0 35117 . . . 4 (((𝐾 ∈ AtLat ∧ ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾) ∧ 𝑢𝐴) ∧ 𝑢 ((𝑃 𝑅) (𝑆 𝑇))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )
4320, 33, 34, 40, 42syl31anc 1479 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )
4443rexlimdv3a 3181 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 ))
4517, 44mpd 15 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  Atomscatm 35070  AtLatcal 35071  HLchlt 35157
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  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-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158
This theorem is referenced by:  ps-2c  35335
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