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Theorem dalem56 36744
Description: Lemma for dath 36752. Analogue of dalem55 36743 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 36672 . . . 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 1125 . . 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 1129 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
87eqcomd 2824 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑍 = 𝑌)
9 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
101, 2, 3, 4, 9dalemswapyzps 36706 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
11 biid 262 . . . 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 262 . . . 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 2818 . . . 4 ((𝑑 𝑆) (𝑐 𝑃)) = ((𝑑 𝑆) (𝑐 𝑃))
18 eqid 2818 . . . 4 ((𝑑 𝑇) (𝑐 𝑄)) = ((𝑑 𝑇) (𝑐 𝑄))
19 eqid 2818 . . . 4 ((𝑑 𝑈) (𝑐 𝑅)) = ((𝑑 𝑈) (𝑐 𝑅))
20 eqid 2818 . . . 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 36743 . . 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 1363 . 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 36640 . . . . . . 7 (𝜑𝐾 ∈ Lat)
25243ad2ant1 1125 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
261dalemkehl 36639 . . . . . . . 8 (𝜑𝐾 ∈ HL)
27263ad2ant1 1125 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
289dalemccea 36699 . . . . . . . 8 (𝜓𝑐𝐴)
29283ad2ant3 1127 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301dalempea 36642 . . . . . . . 8 (𝜑𝑃𝐴)
31303ad2ant1 1125 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
32 eqid 2818 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
3332, 3, 4hlatjcl 36383 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
3427, 29, 31, 33syl3anc 1363 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
359dalemddea 36700 . . . . . . . 8 (𝜓𝑑𝐴)
36353ad2ant3 1127 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
371dalemsea 36645 . . . . . . . 8 (𝜑𝑆𝐴)
38373ad2ant1 1125 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
3932, 3, 4hlatjcl 36383 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
4027, 36, 38, 39syl3anc 1363 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
4132, 13latmcom 17673 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4225, 34, 40, 41syl3anc 1363 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4323, 42syl5eq 2865 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 = ((𝑑 𝑆) (𝑐 𝑃)))
44 dalem54.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
451dalemqea 36643 . . . . . . . 8 (𝜑𝑄𝐴)
46453ad2ant1 1125 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄𝐴)
4732, 3, 4hlatjcl 36383 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑄𝐴) → (𝑐 𝑄) ∈ (Base‘𝐾))
4827, 29, 46, 47syl3anc 1363 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑄) ∈ (Base‘𝐾))
491dalemtea 36646 . . . . . . . 8 (𝜑𝑇𝐴)
50493ad2ant1 1125 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑇𝐴)
5132, 3, 4hlatjcl 36383 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑇𝐴) → (𝑑 𝑇) ∈ (Base‘𝐾))
5227, 36, 50, 51syl3anc 1363 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑇) ∈ (Base‘𝐾))
5332, 13latmcom 17673 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 𝑇) ∈ (Base‘𝐾)) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5425, 48, 52, 53syl3anc 1363 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5544, 54syl5eq 2865 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 = ((𝑑 𝑇) (𝑐 𝑄)))
5643, 55oveq12d 7163 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) = (((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))))
5756oveq1d 7160 . 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 36644 . . . . . . . . . 10 (𝜑𝑅𝐴)
61603ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅𝐴)
6232, 3, 4hlatjcl 36383 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑅𝐴) → (𝑐 𝑅) ∈ (Base‘𝐾))
6327, 29, 61, 62syl3anc 1363 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑅) ∈ (Base‘𝐾))
641dalemuea 36647 . . . . . . . . . 10 (𝜑𝑈𝐴)
65643ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑈𝐴)
6632, 3, 4hlatjcl 36383 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑈𝐴) → (𝑑 𝑈) ∈ (Base‘𝐾))
6727, 36, 65, 66syl3anc 1363 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑈) ∈ (Base‘𝐾))
6832, 13latmcom 17673 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 𝑈) ∈ (Base‘𝐾)) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
6925, 63, 67, 68syl3anc 1363 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
7059, 69syl5eq 2865 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 = ((𝑑 𝑈) (𝑐 𝑅)))
7156, 70oveq12d 7163 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))))
7271, 7oveq12d 7163 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7358, 72syl5eq 2865 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7456, 73oveq12d 7163 . 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 2863 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-lvols 36516
This theorem is referenced by:  dalem57  36745
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