veronesev1lem - Mathbox for Jiamin Zhao

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Theorem veronesev1lem 50788

Description: Lemma for veronesevrowd 50794. Value of the first coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 15-Aug-2026.)

**Hypothesis**
Ref	Expression
veronesevrow.1	⊢ (𝜑 → 𝑃 ∈ (ℝ ↑_m (1...3)))

**Assertion**
Ref	Expression
veronesev1lem	⊢ (𝜑 → ((veronese‘𝑃)‘1) = ((𝑃‘1)↑2))

Proof of Theorem veronesev1lem Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		veronesevrow.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ ↑_m (1...3)))
2	1	veronesevald 50786	. 2 ⊢ (𝜑 → (veronese‘𝑃) = (𝑥 ∈ (1...6) ↦ (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))
3		iftrue 4491	. . . . . . . . . 10 ⊢ (𝑥 = 1 → if(𝑥 = 1, ((𝑃‘1)↑2), 0) = ((𝑃‘1)↑2))
4	3	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑥 = 1 → (if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) = (((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)))
5	4	oveq1d 7431	. . . . . . . 8 ⊢ (𝑥 = 1 → ((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) = ((((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)))
6	5	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 1 → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
7		1ne2 12476	. . . . . . . . . . . . 13 ⊢ 1 ≠ 2
8		neeq1 3019	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑥 ≠ 2 ↔ 1 ≠ 2))
9	7, 8	mpbiri 261	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 ≠ 2)
10	9	neneqd 2962	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ¬ 𝑥 = 2)
11	10	iffalsed 4496	. . . . . . . . . 10 ⊢ (𝑥 = 1 → if(𝑥 = 2, ((𝑃‘2)↑2), 0) = 0)
12	11	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) = (((𝑃‘1)↑2) + 0))
13	12	oveq1d 7431	. . . . . . . 8 ⊢ (𝑥 = 1 → ((((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) = ((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)))
14	13	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 1 → (((((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
15	6, 14	eqtrd 2797	. . . . . 6 ⊢ (𝑥 = 1 → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
16		1ne3 50756	. . . . . . . . . . 11 ⊢ 1 ≠ 3
17		neeq1 3019	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 ≠ 3 ↔ 1 ≠ 3))
18	16, 17	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 ≠ 3)
19	18	neneqd 2962	. . . . . . . . 9 ⊢ (𝑥 = 1 → ¬ 𝑥 = 3)
20	19	iffalsed 4496	. . . . . . . 8 ⊢ (𝑥 = 1 → if(𝑥 = 3, ((𝑃‘3)↑2), 0) = 0)
21	20	oveq2d 7432	. . . . . . 7 ⊢ (𝑥 = 1 → ((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) = ((((𝑃‘1)↑2) + 0) + 0))
22	21	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 1 → (((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
23		1re 11233	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
24		1lt4 12444	. . . . . . . . . . . . 13 ⊢ 1 < 4
25	23, 24	ltneii 11348	. . . . . . . . . . . 12 ⊢ 1 ≠ 4
26		neeq1 3019	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ≠ 4 ↔ 1 ≠ 4))
27	25, 26	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → 𝑥 ≠ 4)
28	27	neneqd 2962	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ¬ 𝑥 = 4)
29	28	iffalsed 4496	. . . . . . . . 9 ⊢ (𝑥 = 1 → if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) = 0)
30	29	oveq1d 7431	. . . . . . . 8 ⊢ (𝑥 = 1 → (if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) = (0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)))
31	30	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 1 → ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)) = ((0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))
32	31	oveq2d 7432	. . . . . 6 ⊢ (𝑥 = 1 → (((((𝑃‘1)↑2) + 0) + 0) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
33	15, 22, 32	3eqtrd 2801	. . . . 5 ⊢ (𝑥 = 1 → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
34		1lt5 12448	. . . . . . . . . . . 12 ⊢ 1 < 5
35	23, 34	ltneii 11348	. . . . . . . . . . 11 ⊢ 1 ≠ 5
36		neeq1 3019	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 ≠ 5 ↔ 1 ≠ 5))
37	35, 36	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 ≠ 5)
38	37	neneqd 2962	. . . . . . . . 9 ⊢ (𝑥 = 1 → ¬ 𝑥 = 5)
39	38	iffalsed 4496	. . . . . . . 8 ⊢ (𝑥 = 1 → if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0) = 0)
40	39	oveq2d 7432	. . . . . . 7 ⊢ (𝑥 = 1 → (0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) = (0 + 0))
41	40	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 1 → ((0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)) = ((0 + 0) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))
42	41	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 1 → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))))
43		1lt6 12453	. . . . . . . . . . 11 ⊢ 1 < 6
44	23, 43	ltneii 11348	. . . . . . . . . 10 ⊢ 1 ≠ 6
45		neeq1 3019	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 ≠ 6 ↔ 1 ≠ 6))
46	44, 45	mpbiri 261	. . . . . . . . 9 ⊢ (𝑥 = 1 → 𝑥 ≠ 6)
47	46	neneqd 2962	. . . . . . . 8 ⊢ (𝑥 = 1 → ¬ 𝑥 = 6)
48	47	iffalsed 4496	. . . . . . 7 ⊢ (𝑥 = 1 → if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0) = 0)
49	48	oveq2d 7432	. . . . . 6 ⊢ (𝑥 = 1 → ((0 + 0) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)) = ((0 + 0) + 0))
50	49	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 1 → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)))
51	33, 42, 50	3eqtrd 2801	. . . 4 ⊢ (𝑥 = 1 → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)))
52	51	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)))
53	1	rr3fv1cld 50762	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃‘1) ∈ ℝ)
54	53	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 1) → (𝑃‘1) ∈ ℝ)
55	54	resqcld 14189	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((𝑃‘1)↑2) ∈ ℝ)
56		0red 11236	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 1) → 0 ∈ ℝ)
57	55, 56	readdcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) ∈ ℝ)
58	57	recnd 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) ∈ ℂ)
59	58	addridd 11435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((((𝑃‘1)↑2) + 0) + 0) = (((𝑃‘1)↑2) + 0))
60	59	oveq1d 7431	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = ((((𝑃‘1)↑2) + 0) + ((0 + 0) + 0)))
61	55	recnd 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((𝑃‘1)↑2) ∈ ℂ)
62	61	addridd 11435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) = ((𝑃‘1)↑2))
63	62	oveq1d 7431	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((((𝑃‘1)↑2) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + ((0 + 0) + 0)))
64	60, 63	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + ((0 + 0) + 0)))
65	56, 56	readdcld 11263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) ∈ ℝ)
66	65	recnd 11262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) ∈ ℂ)
67	66	addridd 11435	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((0 + 0) + 0) = (0 + 0))
68	67	oveq2d 7432	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + (0 + 0)))
69		00id 11410	. . . . . 6 ⊢ (0 + 0) = 0
70	69	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) = 0)
71	70	oveq2d 7432	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + (0 + 0)) = (((𝑃‘1)↑2) + 0))
72	64, 68, 71	3eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + 0))
73	52, 72, 62	3eqtrd 2801	. 2 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((if(𝑥 = 1, ((𝑃‘1)↑2), 0) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0))) = ((𝑃‘1)↑2))
74		1zzd 12650	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
75		6nn 12355	. . . . 5 ⊢ 6 ∈ ℕ
76	75	nnzi 12643	. . . 4 ⊢ 6 ∈ ℤ
77	76	a1i 11	. . 3 ⊢ (𝜑 → 6 ∈ ℤ)
78		1le1 11867	. . . 4 ⊢ 1 ≤ 1
79	78	a1i 11	. . 3 ⊢ (𝜑 → 1 ≤ 1)
80		6re 12356	. . . . 5 ⊢ 6 ∈ ℝ
81	23, 80, 43	ltleii 11358	. . . 4 ⊢ 1 ≤ 6
82	81	a1i 11	. . 3 ⊢ (𝜑 → 1 ≤ 6)
83	74, 77, 74, 79, 82	elfzd 13569	. 2 ⊢ (𝜑 → 1 ∈ (1...6))
84	53	resqcld 14189	. 2 ⊢ (𝜑 → ((𝑃‘1)↑2) ∈ ℝ)
85	2, 73, 83, 84	fvmptd 6998	1 ⊢ (𝜑 → ((veronese‘𝑃)‘1) = ((𝑃‘1)↑2))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ifcif 4485 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 ↑_m cmap 8829 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 ≤ cle 11269 2c2 12320 3c3 12321 4c4 12322 5c5 12323 6c6 12324 ℤcz 12616 ...cfz 13561 ↑cexp 14125 veronesecveronese 50783

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13562 df-seq 14066 df-exp 14126 df-veronese 50784

This theorem is used by: veronesevrowd 50794 veroquadgsumlem 50798

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