Theorem crosspdot0lem 50796

Description: Lemma for crosspdotd 50798. Unfold the curried scalar triple product application into an explicit group sum. (Contributed by Jiamin Zhao, 12-Aug-2026.)

Hypotheses

Ref	Expression
crosspdot0lem.1	⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))
crosspdot0lem.2	⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))
crosspdot0lem.3	⊢ (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))

Assertion

Ref	Expression
crosspdot0lem	⊢ (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝐶,𝑘

Allowed substitution hint:   𝜑(𝑘)

Proof of Theorem crosspdot0lem

Dummy variables 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tripp 50786	. . 3 ⊢ tripp = (𝑠 ∈ (ℝ ↑m (1...3)) ↦ (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))))
2		fveq1 6877	. . . . . . 7 ⊢ (𝑠 = 𝐴 → (𝑠‘𝑘) = (𝐴‘𝑘))
3	2	oveq1d 7428	. . . . . 6 ⊢ (𝑠 = 𝐴 → ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))
4	3	mpteq2dv 5199	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))
5	4	oveq2d 7429	. . . 4 ⊢ (𝑠 = 𝐴 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))
6	5	mpoeq3dv 7492	. . 3 ⊢ (𝑠 = 𝐴 → (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) = (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))))
7		crosspdot0lem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))
8		ovex 7446	. . . . 5 ⊢ (ℝ ↑m (1...3)) ∈ V
9	8, 8	mpoex 8078	. . . 4 ⊢ (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) ∈ V)
11	1, 6, 7, 10	fvmptd3 7010	. 2 ⊢ (𝜑 → (tripp‘𝐴) = (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))))
12		oveq12 7422	. . . . . . 7 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑤⊠𝑧) = (𝐵⊠𝐶))
13	12	fveq1d 6880	. . . . . 6 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑤⊠𝑧)‘𝑘) = ((𝐵⊠𝐶)‘𝑘))
14	13	oveq2d 7429	. . . . 5 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
15	14	mpteq2dv 5199	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
16	15	oveq2d 7429	. . 3 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))
17	16	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))
18		crosspdot0lem.2	. 2 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))
19		crosspdot0lem.3	. 2 ⊢ (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))
20		ovexd 7448	. 2 ⊢ (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) ∈ V)
21	11, 17, 18, 19, 20	ovmpod 7565	1 ⊢ (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   ↑_m cmap 8826  ℝcr 11123  1c1 11125   · cmul 11129  3c3 12320  ...cfz 13561   Σ_g cgsu 17525  ℝ_fldcrefld 21817  ⊠ccrossp 50783  trippctripp 50785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-tripp 50786

This theorem is used by:  crosspdotd  50798