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Theorem dalem54 36730
Description: Lemma for dath 36740. 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 36627 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . 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 36700 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
14 dalem54.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem29 36705 . . 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 36717 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)
18 eqid 2824 . . . 4 (LLines‘𝐾) = (LLines‘𝐾)
195, 6, 18llni2 36516 . . 3 (((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) ∧ 𝐺𝐻) → (𝐺 𝐻) ∈ (LLines‘𝐾))
203, 13, 15, 17, 19syl31anc 1367 . 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 36729 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
231dalemkelat 36628 . . . . . . 7 (𝜑𝐾 ∈ Lat)
24233ad2ant1 1127 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
25 eqid 2824 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2625, 18llnbase 36513 . . . . . . . 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 36710 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2925, 6atbase 36293 . . . . . . . 8 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3125, 5latjcl 17653 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3224, 27, 30, 31syl3anc 1365 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
331, 9dalemyeb 36653 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
34333ad2ant1 1127 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
3525, 4, 8latmle2 17679 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3624, 32, 34, 35syl3anc 1365 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3721, 36eqbrtrid 5097 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 𝑌)
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 36701 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
3925, 6atbase 36293 . . . . . . . 8 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
4013, 39syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
4125, 6atbase 36293 . . . . . . . 8 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
4215, 41syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
4325, 4, 5latjle12 17664 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
4424, 40, 42, 34, 43syl13anc 1366 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
45 simpl 483 . . . . . 6 ((𝐺 𝑌𝐻 𝑌) → 𝐺 𝑌)
4644, 45syl6bir 255 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑌𝐺 𝑌))
4738, 46mtod 199 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝐺 𝐻) 𝑌)
48 nbrne2 5082 . . . 4 ((𝐵 𝑌 ∧ ¬ (𝐺 𝐻) 𝑌) → 𝐵 ≠ (𝐺 𝐻))
4937, 47, 48syl2anc 584 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ≠ (𝐺 𝐻))
5049necomd 3075 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ≠ 𝐵)
51 hlatl 36364 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
523, 51syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
5325, 18llnbase 36513 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5422, 53syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5525, 8latmcl 17654 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
5624, 27, 54, 55syl3anc 1365 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 36728 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 5, 6dalempjqeb 36649 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
59583ad2ant1 1127 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6025, 4, 8latmle1 17678 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
6124, 27, 59, 60syl3anc 1365 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 36727 . . . . . 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 495 . . . . 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 36293 . . . . . . . 8 (𝑐𝐴𝑐 ∈ (Base‘𝐾))
6564anim2i 616 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
66653anim1i 1146 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
67 biid 262 . . . . . . 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 2824 . . . . . . 7 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
69 eqid 2824 . . . . . . 7 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 36677 . . . . . 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 1157 . . . . 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 17654 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7424, 27, 59, 73syl3anc 1365 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7525, 4, 8latlem12 17680 . . . . 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 1366 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
7761, 72, 76mpbi2and 708 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
78 eqid 2824 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7925, 4, 78, 6atlen0 36314 . . 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 1367 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ≠ (0.‘𝐾))
818, 78, 6, 182llnmat 36528 . 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 1372 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  0.cp0 17639  Latclat 17647  Atomscatm 36267  AtLatcal 36268  HLchlt 36354  LLinesclln 36495  LPlanesclpl 36496
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  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 36180  df-ol 36182  df-oml 36183  df-covers 36270  df-ats 36271  df-atl 36302  df-cvlat 36326  df-hlat 36355  df-llines 36502  df-lplanes 36503  df-lvols 36504
This theorem is referenced by:  dalem55  36731  dalem57  36733
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