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Theorem dalem56 34529
Description: Lemma for dath 34537. Analogue of dalem55 34528 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 34457 . . . 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 1080 . . 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 1060 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
87eqcomd 2627 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑍 = 𝑌)
9 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
101, 2, 3, 4, 9dalemswapyzps 34491 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
11 biid 251 . . . 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 251 . . . 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 2621 . . . 4 ((𝑑 𝑆) (𝑐 𝑃)) = ((𝑑 𝑆) (𝑐 𝑃))
18 eqid 2621 . . . 4 ((𝑑 𝑇) (𝑐 𝑄)) = ((𝑑 𝑇) (𝑐 𝑄))
19 eqid 2621 . . . 4 ((𝑑 𝑈) (𝑐 𝑅)) = ((𝑑 𝑈) (𝑐 𝑅))
20 eqid 2621 . . . 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 34528 . . 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 1323 . 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 34425 . . . . . . 7 (𝜑𝐾 ∈ Lat)
25243ad2ant1 1080 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
261dalemkehl 34424 . . . . . . . 8 (𝜑𝐾 ∈ HL)
27263ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
289dalemccea 34484 . . . . . . . 8 (𝜓𝑐𝐴)
29283ad2ant3 1082 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301dalempea 34427 . . . . . . . 8 (𝜑𝑃𝐴)
31303ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
32 eqid 2621 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
3332, 3, 4hlatjcl 34168 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
3427, 29, 31, 33syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
359dalemddea 34485 . . . . . . . 8 (𝜓𝑑𝐴)
36353ad2ant3 1082 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
371dalemsea 34430 . . . . . . . 8 (𝜑𝑆𝐴)
38373ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
3932, 3, 4hlatjcl 34168 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
4027, 36, 38, 39syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
4132, 13latmcom 17007 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4225, 34, 40, 41syl3anc 1323 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4323, 42syl5eq 2667 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 = ((𝑑 𝑆) (𝑐 𝑃)))
44 dalem54.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
451dalemqea 34428 . . . . . . . 8 (𝜑𝑄𝐴)
46453ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄𝐴)
4732, 3, 4hlatjcl 34168 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑄𝐴) → (𝑐 𝑄) ∈ (Base‘𝐾))
4827, 29, 46, 47syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑄) ∈ (Base‘𝐾))
491dalemtea 34431 . . . . . . . 8 (𝜑𝑇𝐴)
50493ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑇𝐴)
5132, 3, 4hlatjcl 34168 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑇𝐴) → (𝑑 𝑇) ∈ (Base‘𝐾))
5227, 36, 50, 51syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑇) ∈ (Base‘𝐾))
5332, 13latmcom 17007 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 𝑇) ∈ (Base‘𝐾)) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5425, 48, 52, 53syl3anc 1323 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5544, 54syl5eq 2667 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 = ((𝑑 𝑇) (𝑐 𝑄)))
5643, 55oveq12d 6628 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) = (((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))))
5756oveq1d 6625 . 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 34429 . . . . . . . . . 10 (𝜑𝑅𝐴)
61603ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅𝐴)
6232, 3, 4hlatjcl 34168 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑅𝐴) → (𝑐 𝑅) ∈ (Base‘𝐾))
6327, 29, 61, 62syl3anc 1323 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑅) ∈ (Base‘𝐾))
641dalemuea 34432 . . . . . . . . . 10 (𝜑𝑈𝐴)
65643ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑈𝐴)
6632, 3, 4hlatjcl 34168 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑈𝐴) → (𝑑 𝑈) ∈ (Base‘𝐾))
6727, 36, 65, 66syl3anc 1323 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑈) ∈ (Base‘𝐾))
6832, 13latmcom 17007 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 𝑈) ∈ (Base‘𝐾)) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
6925, 63, 67, 68syl3anc 1323 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
7059, 69syl5eq 2667 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 = ((𝑑 𝑈) (𝑐 𝑅)))
7156, 70oveq12d 6628 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))))
7271, 7oveq12d 6628 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7358, 72syl5eq 2667 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7456, 73oveq12d 6628 . 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 2665 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  meetcmee 16877  Latclat 16977  Atomscatm 34065  HLchlt 34152  LPlanesclpl 34293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299  df-lplanes 34300  df-lvols 34301
This theorem is referenced by:  dalem57  34530
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