Theorem veronesevald 50804

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 6877	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘1) = (𝑃‘1))
3	2	oveq1d 7428	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1)↑2) = ((𝑃‘1)↑2))
4	3	ifeq1d 4502	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 1, ((𝑞‘1)↑2), 0) = if(𝑘 = 1, ((𝑃‘1)↑2), 0))
5		fveq1 6877	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘2) = (𝑃‘2))
6	5	oveq1d 7428	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2)↑2) = ((𝑃‘2)↑2))
7	6	ifeq1d 4502	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 2, ((𝑞‘2)↑2), 0) = if(𝑘 = 2, ((𝑃‘2)↑2), 0))
8	4, 7	oveq12d 7431	. . . . . 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 6877	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → (𝑞‘3) = (𝑃‘3))
10	9	oveq1d 7428	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3)↑2) = ((𝑃‘3)↑2))
11	10	ifeq1d 4502	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 3, ((𝑞‘3)↑2), 0) = if(𝑘 = 3, ((𝑃‘3)↑2), 0))
12	8, 11	oveq12d 7431	. . . . 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 7431	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1) · (𝑞‘2)) = ((𝑃‘1) · (𝑃‘2)))
14	13	ifeq1d 4502	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) = if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0))
15	5, 9	oveq12d 7431	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2) · (𝑞‘3)) = ((𝑃‘2) · (𝑃‘3)))
16	15	ifeq1d 4502	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0) = if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0))
17	14, 16	oveq12d 7431	. . . . . 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 7431	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3) · (𝑞‘1)) = ((𝑃‘3) · (𝑃‘1)))
19	18	ifeq1d 4502	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0) = if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0))
20	17, 19	oveq12d 7431	. . . . 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 7431	. . . 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 5199	. . 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 50802	. . 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 7446	. . . 4 ⊢ (1...6) ∈ V
25	24	mptex 7222	. . 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 6992	. 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 2145  ifcif 4482   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826  ℝcr 11123  0cc0 11124  1c1 11125   + caddc 11127   · cmul 11129  2c2 12319  3c3 12320  4c4 12321  5c5 12322  6c6 12323  ...cfz 13561  ↑cexp 14125  veronesecveronese 50801

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-veronese 50802

This theorem is used by:  veronesefvcl  50805  veronesev1lem  50806  veronesev2lem  50807  veronesev3lem  50808  veronesev4lem  50809  veronesev5lem  50810  veronesev6lem  50811  veronesevrowd  50812  veronesematrowd  50814