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Theorem dalem56 38158
Description: Lemma for dath 38166. Analogue of dalem55 38157 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 38086 . . . 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 1133 . . 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 1137 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
87eqcomd 2742 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑍 = 𝑌)
9 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
101, 2, 3, 4, 9dalemswapyzps 38120 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
11 biid 260 . . . 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 260 . . . 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 2736 . . . 4 ((𝑑 𝑆) (𝑐 𝑃)) = ((𝑑 𝑆) (𝑐 𝑃))
18 eqid 2736 . . . 4 ((𝑑 𝑇) (𝑐 𝑄)) = ((𝑑 𝑇) (𝑐 𝑄))
19 eqid 2736 . . . 4 ((𝑑 𝑈) (𝑐 𝑅)) = ((𝑑 𝑈) (𝑐 𝑅))
20 eqid 2736 . . . 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 38157 . . 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 1371 . 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 38054 . . . . . . 7 (𝜑𝐾 ∈ Lat)
25243ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
261dalemkehl 38053 . . . . . . . 8 (𝜑𝐾 ∈ HL)
27263ad2ant1 1133 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
289dalemccea 38113 . . . . . . . 8 (𝜓𝑐𝐴)
29283ad2ant3 1135 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301dalempea 38056 . . . . . . . 8 (𝜑𝑃𝐴)
31303ad2ant1 1133 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
32 eqid 2736 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
3332, 3, 4hlatjcl 37796 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
3427, 29, 31, 33syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
359dalemddea 38114 . . . . . . . 8 (𝜓𝑑𝐴)
36353ad2ant3 1135 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
371dalemsea 38059 . . . . . . . 8 (𝜑𝑆𝐴)
38373ad2ant1 1133 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
3932, 3, 4hlatjcl 37796 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
4027, 36, 38, 39syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
4132, 13latmcom 18344 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4225, 34, 40, 41syl3anc 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑑 𝑆) (𝑐 𝑃)))
4323, 42eqtrid 2788 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 = ((𝑑 𝑆) (𝑐 𝑃)))
44 dalem54.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
451dalemqea 38057 . . . . . . . 8 (𝜑𝑄𝐴)
46453ad2ant1 1133 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄𝐴)
4732, 3, 4hlatjcl 37796 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑄𝐴) → (𝑐 𝑄) ∈ (Base‘𝐾))
4827, 29, 46, 47syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑄) ∈ (Base‘𝐾))
491dalemtea 38060 . . . . . . . 8 (𝜑𝑇𝐴)
50493ad2ant1 1133 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑇𝐴)
5132, 3, 4hlatjcl 37796 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑇𝐴) → (𝑑 𝑇) ∈ (Base‘𝐾))
5227, 36, 50, 51syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑇) ∈ (Base‘𝐾))
5332, 13latmcom 18344 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑐 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 𝑇) ∈ (Base‘𝐾)) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5425, 48, 52, 53syl3anc 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑑 𝑇) (𝑐 𝑄)))
5544, 54eqtrid 2788 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 = ((𝑑 𝑇) (𝑐 𝑄)))
5643, 55oveq12d 7371 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) = (((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))))
5756oveq1d 7368 . 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 38058 . . . . . . . . . 10 (𝜑𝑅𝐴)
61603ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅𝐴)
6232, 3, 4hlatjcl 37796 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑅𝐴) → (𝑐 𝑅) ∈ (Base‘𝐾))
6327, 29, 61, 62syl3anc 1371 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑅) ∈ (Base‘𝐾))
641dalemuea 38061 . . . . . . . . . 10 (𝜑𝑈𝐴)
65643ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑈𝐴)
6632, 3, 4hlatjcl 37796 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑈𝐴) → (𝑑 𝑈) ∈ (Base‘𝐾))
6727, 36, 65, 66syl3anc 1371 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑈) ∈ (Base‘𝐾))
6832, 13latmcom 18344 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 𝑈) ∈ (Base‘𝐾)) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
6925, 63, 67, 68syl3anc 1371 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑑 𝑈) (𝑐 𝑅)))
7059, 69eqtrid 2788 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 = ((𝑑 𝑈) (𝑐 𝑅)))
7156, 70oveq12d 7371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))))
7271, 7oveq12d 7371 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑌) = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7358, 72eqtrid 2788 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 = (((((𝑑 𝑆) (𝑐 𝑃)) ((𝑑 𝑇) (𝑐 𝑄))) ((𝑑 𝑈) (𝑐 𝑅))) 𝑍))
7456, 73oveq12d 7371 . 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 2786 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941   class class class wbr 5103  cfv 6493  (class class class)co 7353  Basecbs 17075  lecple 17132  joincjn 18192  meetcmee 18193  Latclat 18312  Atomscatm 37692  HLchlt 37779  LPlanesclpl 37922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-proset 18176  df-poset 18194  df-plt 18211  df-lub 18227  df-glb 18228  df-join 18229  df-meet 18230  df-p0 18306  df-lat 18313  df-clat 18380  df-oposet 37605  df-ol 37607  df-oml 37608  df-covers 37695  df-ats 37696  df-atl 37727  df-cvlat 37751  df-hlat 37780  df-llines 37928  df-lplanes 37929  df-lvols 37930
This theorem is referenced by:  dalem57  38159
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