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Theorem 4atex3 39463
Description: More general version of 4atex 39458 for a line 𝑆 ∨ 𝑇 not necessarily connected to 𝑃 ∨ 𝑄. (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
4that.l ≀ = (leβ€˜πΎ)
4that.j ∨ = (joinβ€˜πΎ)
4that.a 𝐴 = (Atomsβ€˜πΎ)
4that.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
4atex3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇))))
Distinct variable groups:   𝑧,π‘Ÿ,𝐴   𝐻,π‘Ÿ   ∨ ,π‘Ÿ,𝑧   𝐾,π‘Ÿ,𝑧   ≀ ,π‘Ÿ,𝑧   𝑃,π‘Ÿ,𝑧   𝑄,π‘Ÿ,𝑧   𝑆,π‘Ÿ,𝑧   π‘Š,π‘Ÿ,𝑧   𝑇,π‘Ÿ,𝑧   𝑧,𝐻

Proof of Theorem 4atex3
StepHypRef Expression
1 simp1 1133 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simp2 1134 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)))
3 simp31 1206 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑃 β‰  𝑄)
4 simp32l 1295 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑇 ∈ 𝐴)
5 simp33 1208 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
6 4that.l . . . 4 ≀ = (leβ€˜πΎ)
7 4that.j . . . 4 ∨ = (joinβ€˜πΎ)
8 4that.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
9 4that.h . . . 4 𝐻 = (LHypβ€˜πΎ)
106, 7, 8, 94atex2 39459 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 ∈ 𝐴 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
111, 2, 3, 4, 5, 10syl113anc 1379 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
12 simp1l 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ HL)
13 hlcvl 38740 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ CvLat)
1412, 13syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ CvLat)
1514adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ 𝐾 ∈ CvLat)
16 simp23l 1291 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑆 ∈ 𝐴)
1716adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ 𝑆 ∈ 𝐴)
184adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ 𝑇 ∈ 𝐴)
19 simpr 484 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 ∈ 𝐴)
20 simp32r 1296 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑆 β‰  𝑇)
2120adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ 𝑆 β‰  𝑇)
228, 6, 7cvlsupr2 38724 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑆 β‰  𝑇) β†’ ((𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧) ↔ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇))))
2315, 17, 18, 19, 21, 22syl131anc 1380 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ ((𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧) ↔ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇))))
2423anbi2d 628 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑧 ∈ 𝐴) β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)) ↔ (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇)))))
2524rexbidva 3170 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)) ↔ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇)))))
2611, 25mpbid 231 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 β‰  𝑇) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑆 ∧ 𝑧 β‰  𝑇 ∧ 𝑧 ≀ (𝑆 ∨ 𝑇))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  lecple 17211  joincjn 18274  Atomscatm 38644  CvLatclc 38646  HLchlt 38731  LHypclh 39366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-llines 38880  df-lplanes 38881  df-lhyp 39370
This theorem is referenced by:  cdlemg33b0  40083  cdlemg33c0  40084  cdlemg33a  40088
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