Theorem veronesevald 50971

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 6884	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘1) = (𝑃‘1))
3	2	oveq1d 7435	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1)↑2) = ((𝑃‘1)↑2))
4	3	ifeq1d 4501	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 1, ((𝑞‘1)↑2), 0) = if(𝑘 = 1, ((𝑃‘1)↑2), 0))
5		fveq1 6884	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑞‘2) = (𝑃‘2))
6	5	oveq1d 7435	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2)↑2) = ((𝑃‘2)↑2))
7	6	ifeq1d 4501	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 2, ((𝑞‘2)↑2), 0) = if(𝑘 = 2, ((𝑃‘2)↑2), 0))
8	4, 7	oveq12d 7438	. . . . . 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 6884	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → (𝑞‘3) = (𝑃‘3))
10	9	oveq1d 7435	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3)↑2) = ((𝑃‘3)↑2))
11	10	ifeq1d 4501	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 3, ((𝑞‘3)↑2), 0) = if(𝑘 = 3, ((𝑃‘3)↑2), 0))
12	8, 11	oveq12d 7438	. . . . 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 7438	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘1) · (𝑞‘2)) = ((𝑃‘1) · (𝑃‘2)))
14	13	ifeq1d 4501	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) = if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0))
15	5, 9	oveq12d 7438	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((𝑞‘2) · (𝑞‘3)) = ((𝑃‘2) · (𝑃‘3)))
16	15	ifeq1d 4501	. . . . . . 7 ⊢ (𝑞 = 𝑃 → if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0) = if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0))
17	14, 16	oveq12d 7438	. . . . . 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 7438	. . . . . . 7 ⊢ (𝑞 = 𝑃 → ((𝑞‘3) · (𝑞‘1)) = ((𝑃‘3) · (𝑃‘1)))
19	18	ifeq1d 4501	. . . . . 6 ⊢ (𝑞 = 𝑃 → if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0) = if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0))
20	17, 19	oveq12d 7438	. . . . 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 7438	. . . 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 5198	. . 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 50969	. . 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 7453	. . . 4 ⊢ (1...6) ∈ V
25	24	mptex 7229	. . 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 6999	. 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 1569   ∈ wcel 2144  ifcif 4481   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205  2c2 12397  3c3 12398  4c4 12399  5c5 12400  6c6 12401  ...cfz 13639  ↑cexp 14204  veronesecveronese 50968

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-pr 5390

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 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-veronese 50969

This theorem is used by:  veronesefvcl  50972  veronesev1lem  50973  veronesev2lem  50974  veronesev3lem  50975  veronesev4lem  50976  veronesev5lem  50977  veronesev6lem  50978  veronesevrowd  50979  veronesematrowd  50981