P. 508 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50701-50710   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	crosspv3d 50701	Value of the third component of the cross product. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → ((𝐴⊠𝐵)‘3) = (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))))
Theorem	crosspdot0lem 50702*	Lemma for crosspdotd 50704. Unfold the curried scalar triple product application into an explicit group sum. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))
Theorem	crosspdotsumlem 50703*	Lemma for crosspdotd 50704. Expand the group sum over (1...3) into an explicit three-term sum. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3)))))
Theorem	crosspdotd 50704	Value of the scalar triple product, expanded into the standard six-term Sarrus polynomial. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (((((𝐴‘1) · (𝐵‘2)) · (𝐶‘3)) − (((𝐴‘1) · (𝐵‘3)) · (𝐶‘2))) + (((((𝐴‘2) · (𝐵‘3)) · (𝐶‘1)) − (((𝐴‘2) · (𝐵‘1)) · (𝐶‘3))) + ((((𝐴‘3) · (𝐵‘1)) · (𝐶‘2)) − (((𝐴‘3) · (𝐵‘2)) · (𝐶‘1))))))
Theorem	crosspaltd 50705*	Antisymmetry of the cross product: swapping the two vectors negates the result. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐴⊠𝐵) = (𝑘 ∈ (1...3) ↦ -((𝐵⊠𝐴)‘𝑘)))
Theorem	crossp3d 50706*	The vector triple product expansion (BAC-CAB rule): the cross product of 𝑋 with (𝑌⊠𝑍) equals 𝑌 scaled by the dot product of 𝑋 and 𝑍, minus 𝑍 scaled by the dot product of 𝑋 and 𝑌. The dot products are written out as explicit three-term sums of component products, matching the pointwise style of df-crossp 50690 rather than introducing a separate dot product operator. (Contributed by Jiamin Zhao, 12-Aug-2026.)
⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝑍 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝑋⊠(𝑌⊠𝑍)) = (𝑘 ∈ (1...3) ↦ ((((((𝑋‘1) · (𝑍‘1)) + ((𝑋‘2) · (𝑍‘2))) + ((𝑋‘3) · (𝑍‘3))) · (𝑌‘𝑘)) − (((((𝑋‘1) · (𝑌‘1)) + ((𝑋‘2) · (𝑌‘2))) + ((𝑋‘3) · (𝑌‘3))) · (𝑍‘𝑘)))))
21.56  Mathbox for Kunhao Zheng
21.56.1  Weighted AM-GM inequality
Theorem	amgmwlem 50707	Weighted version of amgmlem 27205. (Contributed by Kunhao Zheng, 19-Jun-2021.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ+)    &   ⊢ (𝜑 → 𝑊:𝐴⟶ℝ+)    &   ⊢ (𝜑 → (ℂfld Σg 𝑊) = 1)    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐹 ∘f ↑𝑐𝑊)) ≤ (ℂfld Σg (𝐹 ∘f · 𝑊)))
Theorem	amgmlemALT 50708	Alternate proof of amgmlem 27205 using amgmwlem 50707. (Contributed by Kunhao Zheng, 20-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ+)    ⇒   ⊢ (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (♯‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (♯‘𝐴)))
Theorem	amgmw2d 50709	Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 44982). (Contributed by Kunhao Zheng, 20-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑃 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑄 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑃 + 𝑄) = 1)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
Theorem	young2d 50710	Young's inequality for 𝑛 = 2, a direct application of amgmw2d 50709. (Contributed by Kunhao Zheng, 6-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑃 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑄 ∈ ℝ+)    &   ⊢ (𝜑 → ((1 / 𝑃) + (1 / 𝑄)) = 1)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ≤ (((𝐴↑𝑐𝑃) / 𝑃) + ((𝐵↑𝑐𝑄) / 𝑄)))

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