Theorem crosspdot0lem 51391

Description: Lemma for crosspdotd 51393. 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 51381	. . 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 6915	. . . . . . 7 ⊢ (𝑠 = 𝐴 → (𝑠‘𝑘) = (𝐴‘𝑘))
3	2	oveq1d 7466	. . . . . 6 ⊢ (𝑠 = 𝐴 → ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))
4	3	mpteq2dv 5205	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))
5	4	oveq2d 7467	. . . 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 7530	. . 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 7484	. . . . 5 ⊢ (ℝ ↑m (1...3)) ∈ V
9	8, 8	mpoex 8127	. . . 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 7048	. 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 7460	. . . . . . 7 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑤⊠𝑧) = (𝐵⊠𝐶))
13	12	fveq1d 6918	. . . . . 6 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑤⊠𝑧)‘𝑘) = ((𝐵⊠𝐶)‘𝑘))
14	13	oveq2d 7467	. . . . 5 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
15	14	mpteq2dv 5205	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
16	15	oveq2d 7467	. . 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 7486	. 2 ⊢ (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) ∈ V)
21	11, 17, 18, 19, 20	ovmpod 7604	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 2147  Vcvv 3454   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ↑_m cmap 8885  ℝcr 11238  1c1 11240   · cmul 11244  3c3 12438  ...cfz 13679   Σ_g cgsu 17661  ℝ_fldcrefld 21965  ⊠ccrossp 51378  trippctripp 51380

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-tripp 51381

This theorem is used by:  crosspdotd  51393