Theorem crosspdot0lem 50702

Description: Lemma for crosspdotd 50704. 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 50692	. . 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 6884	. . . . . . 7 ⊢ (𝑠 = 𝐴 → (𝑠‘𝑘) = (𝐴‘𝑘))
3	2	oveq1d 7434	. . . . . 6 ⊢ (𝑠 = 𝐴 → ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))
4	3	mpteq2dv 5207	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))
5	4	oveq2d 7435	. . . 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 7498	. . 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 7452	. . . . 5 ⊢ (ℝ ↑m (1...3)) ∈ V
9	8, 8	mpoex 8082	. . . 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 7017	. 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 7428	. . . . . . 7 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑤⊠𝑧) = (𝐵⊠𝐶))
13	12	fveq1d 6887	. . . . . 6 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑤⊠𝑧)‘𝑘) = ((𝐵⊠𝐶)‘𝑘))
14	13	oveq2d 7435	. . . . 5 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
15	14	mpteq2dv 5207	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
16	15	oveq2d 7435	. . 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 7454	. 2 ⊢ (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) ∈ V)
21	11, 17, 18, 19, 20	ovmpod 7571	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 2146  Vcvv 3457   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421   ↑_m cmap 8830  ℝcr 11114  1c1 11116   · cmul 11120  3c3 12311  ...cfz 13551   Σ_g cgsu 17515  ℝ_fldcrefld 21804  ⊠ccrossp 50689  trippctripp 50691

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-tripp 50692

This theorem is used by:  crosspdotd  50704