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Theorem dalem56 35887
Description: Lemma for dath 35895. Analogue of dalem55 35886 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 35815 . . . 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 1124 . . 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 1128 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
87eqcomd 2784 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑍 = 𝑌)
9 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
101, 2, 3, 4, 9dalemswapyzps 35849 . . 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 2778 . . . 4 ((𝑑 𝑆) (𝑐 𝑃)) = ((𝑑 𝑆) (𝑐 𝑃))
18 eqid 2778 . . . 4 ((𝑑 𝑇) (𝑐 𝑄)) = ((𝑑 𝑇) (𝑐 𝑄))
19 eqid 2778 . . . 4 ((𝑑 𝑈) (𝑐 𝑅)) = ((𝑑 𝑈) (𝑐 𝑅))
20 eqid 2778 . . . 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 35886 . . 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 1439 . 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 35783 . . . . . . 7 (𝜑𝐾 ∈ Lat)
25243ad2ant1 1124 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
261dalemkehl 35782 . . . . . . . 8 (𝜑𝐾 ∈ HL)
27263ad2ant1 1124 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
289dalemccea 35842 . . . . . . . 8 (𝜓𝑐𝐴)
29283ad2ant3 1126 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301dalempea 35785 . . . . . . . 8 (𝜑𝑃𝐴)
31303ad2ant1 1124 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
32 eqid 2778 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
3332, 3, 4hlatjcl 35526 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
3427, 29, 31, 33syl3anc 1439 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
359dalemddea 35843 . . . . . . . 8 (𝜓𝑑𝐴)
36353ad2ant3 1126 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
371dalemsea 35788 . . . . . . . 8 (𝜑𝑆𝐴)
38373ad2ant1 1124 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
3932, 3, 4hlatjcl 35526 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
4027, 36, 38, 39syl3anc 1439 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
4132, 13latmcom 17465 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4225, 34, 40, 41syl3anc 1439 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4323, 42syl5eq 2826 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 = ((𝑑 𝑆) (𝑐 𝑃)))
44 dalem54.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
451dalemqea 35786 . . . . . . . 8 (𝜑𝑄𝐴)
46453ad2ant1 1124 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄𝐴)
4732, 3, 4hlatjcl 35526 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑄𝐴) → (𝑐 𝑄) ∈ (Base‘𝐾))
4827, 29, 46, 47syl3anc 1439 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑄) ∈ (Base‘𝐾))
491dalemtea 35789 . . . . . . . 8 (𝜑𝑇𝐴)
50493ad2ant1 1124 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑇𝐴)
5132, 3, 4hlatjcl 35526 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑇𝐴) → (𝑑 𝑇) ∈ (Base‘𝐾))
5227, 36, 50, 51syl3anc 1439 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑇) ∈ (Base‘𝐾))
5332, 13latmcom 17465 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 𝑇) ∈ (Base‘𝐾)) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5425, 48, 52, 53syl3anc 1439 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5544, 54syl5eq 2826 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 = ((𝑑 𝑇) (𝑐 𝑄)))
5643, 55oveq12d 6942 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) = (((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))))
5756oveq1d 6939 . 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 35787 . . . . . . . . . 10 (𝜑𝑅𝐴)
61603ad2ant1 1124 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅𝐴)
6232, 3, 4hlatjcl 35526 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑅𝐴) → (𝑐 𝑅) ∈ (Base‘𝐾))
6327, 29, 61, 62syl3anc 1439 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑅) ∈ (Base‘𝐾))
641dalemuea 35790 . . . . . . . . . 10 (𝜑𝑈𝐴)
65643ad2ant1 1124 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑈𝐴)
6632, 3, 4hlatjcl 35526 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑈𝐴) → (𝑑 𝑈) ∈ (Base‘𝐾))
6727, 36, 65, 66syl3anc 1439 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑈) ∈ (Base‘𝐾))
6832, 13latmcom 17465 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 𝑈) ∈ (Base‘𝐾)) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
6925, 63, 67, 68syl3anc 1439 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
7059, 69syl5eq 2826 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 = ((𝑑 𝑈) (𝑐 𝑅)))
7156, 70oveq12d 6942 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))))
7271, 7oveq12d 6942 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7358, 72syl5eq 2826 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7456, 73oveq12d 6942 . 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 2824 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 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4888  cfv 6137  (class class class)co 6924  Basecbs 16259  lecple 16349  joincjn 17334  meetcmee 17335  Latclat 17435  Atomscatm 35422  HLchlt 35509  LPlanesclpl 35651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-proset 17318  df-poset 17336  df-plt 17348  df-lub 17364  df-glb 17365  df-join 17366  df-meet 17367  df-p0 17429  df-lat 17436  df-clat 17498  df-oposet 35335  df-ol 35337  df-oml 35338  df-covers 35425  df-ats 35426  df-atl 35457  df-cvlat 35481  df-hlat 35510  df-llines 35657  df-lplanes 35658  df-lvols 35659
This theorem is referenced by:  dalem57  35888
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