Theorem crosspdot0i 50664

Description: Unfold the curried scalar triple product application into an explicit group sum. (A helper for crosspdoti 50666.) (Contributed by Jiamin Zhao, 1-Aug-2026.)

Hypotheses

Ref	Expression
crosspdot0i.1	⊢ 𝐴 ∈ (ℝ ↑m (1...3))
crosspdot0i.2	⊢ 𝐵 ∈ (ℝ ↑m (1...3))
crosspdot0i.3	⊢ 𝐶 ∈ (ℝ ↑m (1...3))

Assertion

Ref	Expression
crosspdot0i	⊢ (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))

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

Proof of Theorem crosspdot0i

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

Step	Hyp	Ref	Expression
1		crosspdot0i.1	. . . 4 ⊢ 𝐴 ∈ (ℝ ↑m (1...3))
2		ovex 7443	. . . . 5 ⊢ (ℝ ↑m (1...3)) ∈ V
3	2, 2	mpoex 8072	. . . 4 ⊢ (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) ∈ V
4		fveq1 6880	. . . . . . . . 9 ⊢ (𝑠 = 𝐴 → (𝑠‘𝑘) = (𝐴‘𝑘))
5	4	oveq1d 7425	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))
6	5	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑠 = 𝐴 → (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))
7	6	oveq2d 7426	. . . . . 6 ⊢ (𝑠 = 𝐴 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))
8	7	mpoeq3dv 7489	. . . . 5 ⊢ (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))))))
9		df-tripp 50654	. . . . 5 ⊢ 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))))))
10	8, 9	fvmptg 6987	. . . 4 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) ∈ V) → (tripp‘𝐴) = (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))))
11	1, 3, 10	mp2an 704	. . 3 ⊢ (tripp‘𝐴) = (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))
12	11	oveqi 7423	. 2 ⊢ (𝐵(tripp‘𝐴)𝐶) = (𝐵(𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))𝐶)
13		eqidd 2764	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))) = (𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))))
14		simprl 782	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → 𝑤 = 𝐵)
15		simprr 784	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → 𝑧 = 𝐶)
16	14, 15	oveq12d 7428	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → (𝑤⊠𝑧) = (𝐵⊠𝐶))
17	16	fveq1d 6883	. . . . . . 7 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → ((𝑤⊠𝑧)‘𝑘) = ((𝐵⊠𝐶)‘𝑘))
18	17	oveq2d 7426	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
19	18	mpteq2dv 5205	. . . . 5 ⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ (𝑤 = 𝐵 ∧ 𝑧 = 𝐶)) → (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
20	19	oveq2d 7426	. . . 4 ⊢ ((𝐴 ∈ (ℝ ↑m (1...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)))))
21		crosspdot0i.2	. . . . 5 ⊢ 𝐵 ∈ (ℝ ↑m (1...3))
22	21	a1i 11	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → 𝐵 ∈ (ℝ ↑m (1...3)))
23		crosspdot0i.3	. . . . 5 ⊢ 𝐶 ∈ (ℝ ↑m (1...3))
24	23	a1i 11	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → 𝐶 ∈ (ℝ ↑m (1...3)))
25		ovexd 7445	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) ∈ V)
26	13, 20, 22, 24, 25	ovmpod 7562	. . 3 ⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → (𝐵(𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))
27	1, 26	ax-mp 5	. 2 ⊢ (𝐵(𝑤 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))))𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
28	12, 27	eqtri 2786	1 ⊢ (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820  ℝcr 11094  1c1 11096   · cmul 11100  3c3 12291  ...cfz 13530   Σ_g cgsu 17488  ℝ_fldcrefld 21754  ⊠ccrossp 50651  trippctripp 50653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-tripp 50654

This theorem is referenced by:  crosspdoti  50666