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Theorem ps-2b 36775
Description: Variation of projective geometry axiom ps-2 36771. (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 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝐾 ∈ HL)
2 simp12 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑃𝐴)
3 simp13 1202 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑄𝐴)
4 simp21 1203 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑅𝐴)
52, 3, 43jca 1125 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
6 simp22 1204 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝐴)
7 simp23 1205 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑇𝐴)
86, 7jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (𝑆𝐴𝑇𝐴))
9 simp31 1206 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ¬ 𝑃 (𝑄 𝑅))
10 simp32 1207 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → 𝑆𝑇)
119, 10jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇))
12 simp33 1208 . . 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 36771 . . 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 1375 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
18 simp111 1299 . . . . 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 36653 . . . . 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 36657 . . . . 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 1300 . . . . . 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 1302 . . . . . 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 2798 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2524, 14, 15hlatjcl 36660 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
2618, 22, 23, 25syl3anc 1368 . . . . 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 1303 . . . . . 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 1304 . . . . . 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 36660 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
3018, 27, 28, 29syl3anc 1368 . . . . 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 17654 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ (Base‘𝐾))
3321, 26, 30, 32syl3anc 1368 . . . 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 1134 . . . 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 1135 . . . . 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 36582 . . . . . . 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 17680 . . . . . 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 1369 . . . . 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 235 . . . 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 36603 . . . 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 1370 . . 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 3245 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  wrex 3107   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  0.cp0 17639  Latclat 17647  Atomscatm 36556  AtLatcal 36557  HLchlt 36643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36469  df-ol 36471  df-oml 36472  df-covers 36559  df-ats 36560  df-atl 36591  df-cvlat 36615  df-hlat 36644
This theorem is referenced by:  ps-2c  36821
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