Theorem veronesevald 51249

Description: Value of the Veronese map at a point, expressed as a maps-to function on the six coordinates. (Contributed by Jiamin Zhao, 14-Aug-2026.)

Hypothesis

Ref	Expression
veroneseval.1	⊢ (𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))

Assertion

Ref	Expression
veronesevald	⊢ (𝜑 → (veronese‘𝑃) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))

Distinct variable group:   𝑃,𝑘

Allowed substitution hint:   𝜑(𝑘)

Proof of Theorem veronesevald

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		veroneseval.1	. 2 ⊢ (𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))
2		fveq1 6904	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘1) = (𝑃‘1))
3	2	oveq1d 7455	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1)↑2) = ((𝑃‘1)↑2))
4	3	ifeq1d 4505	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 1, ((𝑞‘1)↑2), 0) = if(𝑘 = 1, ((𝑃‘1)↑2), 0))
5		fveq1 6904	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘2) = (𝑃‘2))
6	5	oveq1d 7455	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2)↑2) = ((𝑃‘2)↑2))
7	6	ifeq1d 4505	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 2, ((𝑞‘2)↑2), 0) = if(𝑘 = 2, ((𝑃‘2)↑2), 0))
8	4, 7	oveq12d 7458	. . . . . 6 ⊢ (𝑞 = 𝑃 → (if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) = (if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)))
9		fveq1 6904	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → (𝑞‘3) = (𝑃‘3))
10	9	oveq1d 7455	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3)↑2) = ((𝑃‘3)↑2))
11	10	ifeq1d 4505	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 3, ((𝑞‘3)↑2), 0) = if(𝑘 = 3, ((𝑃‘3)↑2), 0))
12	8, 11	oveq12d 7458	. . . . 5 ⊢ (𝑞 = 𝑃 → ((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) = ((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)))
13	2, 5	oveq12d 7458	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1) · (𝑞‘2)) = ((𝑃‘1) · (𝑃‘2)))
14	13	ifeq1d 4505	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) = if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0))
15	5, 9	oveq12d 7458	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2) · (𝑞‘3)) = ((𝑃‘2) · (𝑃‘3)))
16	15	ifeq1d 4505	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0) = if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0))
17	14, 16	oveq12d 7458	. . . . . 6 ⊢ (𝑞 = 𝑃 → (if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) = (if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)))
18	9, 2	oveq12d 7458	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3) · (𝑞‘1)) = ((𝑃‘3) · (𝑃‘1)))
19	18	ifeq1d 4505	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0) = if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0))
20	17, 19	oveq12d 7458	. . . . 5 ⊢ (𝑞 = 𝑃 → ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0)) = ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))
21	12, 20	oveq12d 7458	. . . 4 ⊢ (𝑞 = 𝑃 → (((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0))) = (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
22	21	mpteq2dv 5202	. . 3 ⊢ (𝑞 = 𝑃 → (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0)))) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))
23		df-veronese 51247	. . 3 ⊢ veronese = (𝑞 ∈ (ℝ ↑m (1...3)) ↦ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0)))))
24		ovex 7473	. . . 4 ⊢ (1...6) ∈ V
25	24	mptex 7249	. . 3 ⊢ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0)))) ∈ V
26	22, 23, 25	fvmpt3i 7019	. 2 ⊢ (𝑃 ∈ (ℝ ↑m (1...3)) → (veronese‘𝑃) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))
27	1, 26	syl 18	1 ⊢ (𝜑 → (veronese‘𝑃) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ifcif 4485   ↦ cmpt 5189  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   · cmul 11229  2c2 12421  3c3 12422  4c4 12423  5c5 12424  6c6 12425  ...cfz 13663  ↑cexp 14228  veronesecveronese 51246

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 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

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-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  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 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-veronese 51247

This theorem is used by:  veronesefvcl  51250  veronesev1lem  51251  veronesev2lem  51252  veronesev3lem  51253  veronesev4lem  51254  veronesev5lem  51255  veronesev6lem  51256  veronesevrowd  51257  veronesematrowd  51259