Theorem crosspdot0lem 51145

Description: Lemma for crosspdotd 51147. 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 51135	. . 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 6887	. . . . . . 7 ⊢ (𝑠 = 𝐴 → (𝑠‘𝑘) = (𝐴‘𝑘))
3	2	oveq1d 7437	. . . . . 6 ⊢ (𝑠 = 𝐴 → ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)))
4	3	mpteq2dv 5198	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑘 ∈ (1...3) ↦ ((𝑠‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))))
5	4	oveq2d 7438	. . . 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 7501	. . 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 7455	. . . . 5 ⊢ (ℝ ↑m (1...3)) ∈ V
9	8, 8	mpoex 8094	. . . 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 7020	. 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 7431	. . . . . . 7 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑤⊠𝑧) = (𝐵⊠𝐶))
13	12	fveq1d 6890	. . . . . 6 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑤⊠𝑧)‘𝑘) = ((𝐵⊠𝐶)‘𝑘))
14	13	oveq2d 7438	. . . . 5 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘)) = ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
15	14	mpteq2dv 5198	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝑤⊠𝑧)‘𝑘))) = (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
16	15	oveq2d 7438	. . 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 7457	. 2 ⊢ (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) ∈ V)
21	11, 17, 18, 19, 20	ovmpod 7574	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 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424   ↑_m cmap 8852  ℝcr 11204  1c1 11206   · cmul 11210  3c3 12403  ...cfz 13644   Σ_g cgsu 17618  ℝ_fldcrefld 21917  ⊠ccrossp 51132  trippctripp 51134

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-tripp 51135

This theorem is used by:  crosspdotd  51147