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Theorem dalem54 39425
Description: Lemma for dath 39435. 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 39322 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1130 . 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 39395 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
14 dalem54.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem29 39400 . . 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 39412 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)
18 eqid 2726 . . . 4 (LLines‘𝐾) = (LLines‘𝐾)
195, 6, 18llni2 39211 . . 3 (((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) ∧ 𝐺𝐻) → (𝐺 𝐻) ∈ (LLines‘𝐾))
203, 13, 15, 17, 19syl31anc 1370 . 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 39424 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
231dalemkelat 39323 . . . . . . 7 (𝜑𝐾 ∈ Lat)
24233ad2ant1 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
25 eqid 2726 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2625, 18llnbase 39208 . . . . . . . 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 39405 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2925, 6atbase 38987 . . . . . . . 8 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3125, 5latjcl 18464 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3224, 27, 30, 31syl3anc 1368 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
331, 9dalemyeb 39348 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
34333ad2ant1 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
3525, 4, 8latmle2 18490 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3624, 32, 34, 35syl3anc 1368 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) 𝑌)
3721, 36eqbrtrid 5188 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 𝑌)
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 39396 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
3925, 6atbase 38987 . . . . . . . 8 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
4013, 39syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
4125, 6atbase 38987 . . . . . . . 8 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
4215, 41syl 17 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
4325, 4, 5latjle12 18475 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
4424, 40, 42, 34, 43syl13anc 1369 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝑌𝐻 𝑌) ↔ (𝐺 𝐻) 𝑌))
45 simpl 481 . . . . . 6 ((𝐺 𝑌𝐻 𝑌) → 𝐺 𝑌)
4644, 45biimtrrdi 253 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑌𝐺 𝑌))
4738, 46mtod 197 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝐺 𝐻) 𝑌)
48 nbrne2 5173 . . . 4 ((𝐵 𝑌 ∧ ¬ (𝐺 𝐻) 𝑌) → 𝐵 ≠ (𝐺 𝐻))
4937, 47, 48syl2anc 582 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ≠ (𝐺 𝐻))
5049necomd 2986 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ≠ 𝐵)
51 hlatl 39058 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
523, 51syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
5325, 18llnbase 39208 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5422, 53syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5525, 8latmcl 18465 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
5624, 27, 54, 55syl3anc 1368 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 39423 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 5, 6dalempjqeb 39344 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
59583ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6025, 4, 8latmle1 18489 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
6124, 27, 59, 60syl3anc 1368 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 39422 . . . . . 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 493 . . . . 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 38987 . . . . . . . 8 (𝑐𝐴𝑐 ∈ (Base‘𝐾))
6564anim2i 615 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
66653anim1i 1149 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
67 biid 260 . . . . . . 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 2726 . . . . . . 7 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
69 eqid 2726 . . . . . . 7 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 39372 . . . . . 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 1160 . . . . 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 18465 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7424, 27, 59, 73syl3anc 1368 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
7525, 4, 8latlem12 18491 . . . . 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 1369 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
7761, 72, 76mpbi2and 710 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
78 eqid 2726 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7925, 4, 78, 6atlen0 39008 . . 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 1370 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ≠ (0.‘𝐾))
818, 78, 6, 182llnmat 39223 . 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 1375 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5153  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  0.cp0 18448  Latclat 18456  Atomscatm 38961  AtLatcal 38962  HLchlt 39048  LLinesclln 39190  LPlanesclpl 39191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199
This theorem is referenced by:  dalem55  39426  dalem57  39428
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