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Theorem dalem54 35796
Description: Lemma for dath 35806. Line 𝐺𝐻 intersects the auxiliary axis of perspectivity 𝐵. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem54.m = (meet‘𝐾)
dalem54.o 𝑂 = (LPlanes‘𝐾)
dalem54.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem54.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem54.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem54.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem54.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem54.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem54 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)

Proof of Theorem dalem54
StepHypRef Expression
1 dalem.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
21dalemkehl 35693 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1167 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.l . . . 4 = (le‘𝐾)
5 dalem.j . . . 4 = (join‘𝐾)
6 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
7 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
8 dalem54.m . . . 4 = (meet‘𝐾)
9 dalem54.o . . . 4 𝑂 = (LPlanes‘𝐾)
10 dalem54.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
11 dalem54.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
12 dalem54.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
131, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem23 35766 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
14 dalem54.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem29 35771 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
16 dalem54.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
171, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem41 35783 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)
18 eqid 2825 . . . 4 (LLines‘𝐾) = (LLines‘𝐾)
195, 6, 18llni2 35582 . . 3 (((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) ∧ 𝐺𝐻) → (𝐺 𝐻) ∈ (LLines‘𝐾))
203, 13, 15, 17, 19syl31anc 1496 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (LLines‘𝐾))
21 dalem54.b1 . . 3 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
221, 4, 5, 6, 7, 8, 18, 9, 10, 11, 12, 14, 16, 21dalem53 35795 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
231dalemkelat 35694 . . . . . . 7 (𝜑𝐾 ∈ Lat)
24233ad2ant1 1167 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
25 eqid 2825 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2625, 18llnbase 35579 . . . . . . . 8 ((𝐺 𝐻) ∈ (LLines‘𝐾) → (𝐺 𝐻) ∈ (Base‘𝐾))
2720, 26syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
281, 4, 5, 6, 7, 8, 9, 10, 11, 16dalem34 35776 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2925, 6atbase 35359 . . . . . . . 8 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3125, 5latjcl 17411 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3224, 27, 30, 31syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
331, 9dalemyeb 35719 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
34333ad2ant1 1167 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
3525, 4, 8latmle2 17437 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3624, 32, 34, 35syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3721, 36syl5eqbr 4910 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 𝑌)
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 35767 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
3925, 6atbase 35359 . . . . . . . 8 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
4013, 39syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
4125, 6atbase 35359 . . . . . . . 8 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
4215, 41syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
4325, 4, 5latjle12 17422 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
4424, 40, 42, 34, 43syl13anc 1495 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
45 simpl 476 . . . . . 6 ((𝐺 𝑌𝐻 𝑌) → 𝐺 𝑌)
4644, 45syl6bir 246 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑌𝐺 𝑌))
4738, 46mtod 190 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝐺 𝐻) 𝑌)
48 nbrne2 4895 . . . 4 ((𝐵 𝑌 ∧ ¬ (𝐺 𝐻) 𝑌) → 𝐵 ≠ (𝐺 𝐻))
4937, 47, 48syl2anc 579 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ≠ (𝐺 𝐻))
5049necomd 3054 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ≠ 𝐵)
51 hlatl 35430 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
523, 51syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
5325, 18llnbase 35579 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5422, 53syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5525, 8latmcl 17412 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
5624, 27, 54, 55syl3anc 1494 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 35794 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 5, 6dalempjqeb 35715 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
59583ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6025, 4, 8latmle1 17436 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
6124, 27, 59, 60syl3anc 1494 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 35793 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) ∧ ((𝐺 𝐻) 𝐼) ≠ 𝑌))
6362simpld 490 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))))
6425, 6atbase 35359 . . . . . . . 8 (𝑐𝐴𝑐 ∈ (Base‘𝐾))
6564anim2i 610 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
66653anim1i 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
67 biid 253 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))))
68 eqid 2825 . . . . . . 7 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
69 eqid 2825 . . . . . . 7 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 35743 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
7166, 70syl3an1 1206 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
7263, 71syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
7325, 8latmcl 17412 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7424, 27, 59, 73syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7525, 4, 8latlem12 17438 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
7624, 74, 27, 54, 75syl13anc 1495 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
7761, 72, 76mpbi2and 703 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
78 eqid 2825 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7925, 4, 78, 6atlen0 35380 . . 3 (((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴) ∧ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)) → ((𝐺 𝐻) 𝐵) ≠ (0.‘𝐾))
8052, 56, 57, 77, 79syl31anc 1496 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ≠ (0.‘𝐾))
818, 78, 6, 182llnmat 35594 . 2 (((𝐾 ∈ HL ∧ (𝐺 𝐻) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) ∧ ((𝐺 𝐻) ≠ 𝐵 ∧ ((𝐺 𝐻) 𝐵) ≠ (0.‘𝐾))) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
823, 20, 22, 50, 80, 81syl32anc 1501 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  0.cp0 17397  Latclat 17405  Atomscatm 35333  AtLatcal 35334  HLchlt 35420  LLinesclln 35561  LPlanesclpl 35562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35246  df-ol 35248  df-oml 35249  df-covers 35336  df-ats 35337  df-atl 35368  df-cvlat 35392  df-hlat 35421  df-llines 35568  df-lplanes 35569  df-lvols 35570
This theorem is referenced by:  dalem55  35797  dalem57  35799
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