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Theorem dalem56 35802
Description: Lemma for dath 35810. Analogue of dalem55 35801 for line 𝑆𝑇. (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
dalem56 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))

Proof of Theorem dalem56
StepHypRef Expression
1 dalem.ph . . . . 5 (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)))))
2 dalem.l . . . . 5 = (le‘𝐾)
3 dalem.j . . . . 5 = (join‘𝐾)
4 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4dalemswapyz 35730 . . . 4 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
653ad2ant1 1167 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
7 simp2 1171 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
87eqcomd 2831 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑍 = 𝑌)
9 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
101, 2, 3, 4, 9dalemswapyzps 35764 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
11 biid 253 . . . 4 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
12 biid 253 . . . 4 (((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))) ↔ ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
13 dalem54.m . . . 4 = (meet‘𝐾)
14 dalem54.o . . . 4 𝑂 = (LPlanes‘𝐾)
15 dalem54.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
16 dalem54.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
17 eqid 2825 . . . 4 ((𝑑 𝑆) (𝑐 𝑃)) = ((𝑑 𝑆) (𝑐 𝑃))
18 eqid 2825 . . . 4 ((𝑑 𝑇) (𝑐 𝑄)) = ((𝑑 𝑇) (𝑐 𝑄))
19 eqid 2825 . . . 4 ((𝑑 𝑈) (𝑐 𝑅)) = ((𝑑 𝑈) (𝑐 𝑅))
20 eqid 2825 . . . 4 (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍)
2111, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 19, 20dalem55 35801 . . 3 (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))) ∧ 𝑍 = 𝑌 ∧ ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐)))) → ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (𝑆 𝑇)) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍)))
226, 8, 10, 21syl3anc 1494 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (𝑆 𝑇)) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍)))
23 dalem54.g . . . . 5 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241dalemkelat 35698 . . . . . . 7 (𝜑𝐾 ∈ Lat)
25243ad2ant1 1167 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
261dalemkehl 35697 . . . . . . . 8 (𝜑𝐾 ∈ HL)
27263ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
289dalemccea 35757 . . . . . . . 8 (𝜓𝑐𝐴)
29283ad2ant3 1169 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301dalempea 35700 . . . . . . . 8 (𝜑𝑃𝐴)
31303ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
32 eqid 2825 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
3332, 3, 4hlatjcl 35441 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
3427, 29, 31, 33syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
359dalemddea 35758 . . . . . . . 8 (𝜓𝑑𝐴)
36353ad2ant3 1169 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
371dalemsea 35703 . . . . . . . 8 (𝜑𝑆𝐴)
38373ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
3932, 3, 4hlatjcl 35441 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
4027, 36, 38, 39syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
4132, 13latmcom 17435 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4225, 34, 40, 41syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4323, 42syl5eq 2873 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 = ((𝑑 𝑆) (𝑐 𝑃)))
44 dalem54.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
451dalemqea 35701 . . . . . . . 8 (𝜑𝑄𝐴)
46453ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄𝐴)
4732, 3, 4hlatjcl 35441 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑄𝐴) → (𝑐 𝑄) ∈ (Base‘𝐾))
4827, 29, 46, 47syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑄) ∈ (Base‘𝐾))
491dalemtea 35704 . . . . . . . 8 (𝜑𝑇𝐴)
50493ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑇𝐴)
5132, 3, 4hlatjcl 35441 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑇𝐴) → (𝑑 𝑇) ∈ (Base‘𝐾))
5227, 36, 50, 51syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑇) ∈ (Base‘𝐾))
5332, 13latmcom 17435 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 𝑇) ∈ (Base‘𝐾)) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5425, 48, 52, 53syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5544, 54syl5eq 2873 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 = ((𝑑 𝑇) (𝑐 𝑄)))
5643, 55oveq12d 6928 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) = (((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))))
5756oveq1d 6925 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (𝑆 𝑇)))
58 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
59 dalem54.i . . . . . . 7 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
601dalemrea 35702 . . . . . . . . . 10 (𝜑𝑅𝐴)
61603ad2ant1 1167 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅𝐴)
6232, 3, 4hlatjcl 35441 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑅𝐴) → (𝑐 𝑅) ∈ (Base‘𝐾))
6327, 29, 61, 62syl3anc 1494 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑅) ∈ (Base‘𝐾))
641dalemuea 35705 . . . . . . . . . 10 (𝜑𝑈𝐴)
65643ad2ant1 1167 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑈𝐴)
6632, 3, 4hlatjcl 35441 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑈𝐴) → (𝑑 𝑈) ∈ (Base‘𝐾))
6727, 36, 65, 66syl3anc 1494 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑈) ∈ (Base‘𝐾))
6832, 13latmcom 17435 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 𝑈) ∈ (Base‘𝐾)) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
6925, 63, 67, 68syl3anc 1494 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
7059, 69syl5eq 2873 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 = ((𝑑 𝑈) (𝑐 𝑅)))
7156, 70oveq12d 6928 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))))
7271, 7oveq12d 6928 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7358, 72syl5eq 2873 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7456, 73oveq12d 6928 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍)))
7522, 57, 743eqtr4d 2871 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
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  Latclat 17405  Atomscatm 35337  HLchlt 35424  LPlanesclpl 35566
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 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573  df-lvols 35574
This theorem is referenced by:  dalem57  35803
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