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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 1246 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝐾 ∈ HL)
2 simp12 1247 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃𝐴)
3 simp13 1248 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑄𝐴)
4 simp21 1249 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑅𝐴)
52, 3, 43jca 1123 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
6 simp22 1250 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝐴)
7 simp23 1251 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑇𝐴)
86, 7jca 555 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆𝐴𝑇𝐴))
9 simp31 1252 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ¬ 𝑃 (𝑄 𝑅))
10 simp32 1253 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝑇)
119, 10jca 555 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇))
12 simp33 1254 . . 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 1485 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
18 simp111 1387 . . . . 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 35168 . . . . 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)
21 hllat 35171 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2218, 21syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝐾 ∈ Lat)
23 simp112 1388 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑃𝐴)
24 simp121 1390 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑅𝐴)
25 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2625, 14, 15hlatjcl 35174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
2718, 23, 24, 26syl3anc 1477 . . . . 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))
28 simp122 1391 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑆𝐴)
29 simp123 1392 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑇𝐴)
3025, 14, 15hlatjcl 35174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
3118, 28, 29, 30syl3anc 1477 . . . . 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))
32 ps-2b.m . . . . . 6 = (meet‘𝐾)
3325, 32latmcl 17273 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾))
3422, 27, 31, 33syl3anc 1477 . . . 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))
35 simp2 1132 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) ∧ 𝑢𝐴 ∧ (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇))) → 𝑢𝐴)
36 simp3 1133 . . . . 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)))
3725, 15atbase 35097 . . . . . . 7 (𝑢𝐴𝑢 ∈ (Base‘𝐾))
3835, 37syl 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))
3925, 13, 32latlem12 17299 . . . . . 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))))
4022, 38, 27, 31, 39syl13anc 1479 . . . . 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))))
4136, 40mpbid 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)))
42 ps-2b.z . . . . 5 0 = (0.‘𝐾)
4325, 13, 42, 15atlen0 35118 . . . 4 (((𝐾 ∈ AtLat ∧ ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾) ∧ 𝑢𝐴) ∧ 𝑢 ((𝑃 𝑅) (𝑆 𝑇))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )
4420, 34, 35, 41, 43syl31anc 1480 . . 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 )
4544rexlimdv3a 3171 . 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 ))
4617, 45mpd 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  0.cp0 17258  Latclat 17266  Atomscatm 35071  AtLatcal 35072  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-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159
This theorem is referenced by:  ps-2c  35335
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