veronesev1lem - Mathbox for Jiamin Zhao

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Theorem veronesev1lem 50973

Description: Lemma for veronesevrowd 50979. 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 50971	. 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 4487	. . . . . . . . . 10 ⊢ (𝑥 = 1 → if(𝑥 = 1, ((𝑃‘1)↑2), 0) = ((𝑃‘1)↑2))
4	3	oveq1d 7435	. . . . . . . . 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 7435	. . . . . . . 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 7435	. . . . . . 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 12553	. . . . . . . . . . . . 13 ⊢ 1 ≠ 2
8		neeq1 3017	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑥 ≠ 2 ↔ 1 ≠ 2))
9	7, 8	mpbiri 261	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 ≠ 2)
10	9	neneqd 2960	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ¬ 𝑥 = 2)
11	10	iffalsed 4492	. . . . . . . . . 10 ⊢ (𝑥 = 1 → if(𝑥 = 2, ((𝑃‘2)↑2), 0) = 0)
12	11	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝑃‘1)↑2) + if(𝑥 = 2, ((𝑃‘2)↑2), 0)) = (((𝑃‘1)↑2) + 0))
13	12	oveq1d 7435	. . . . . . . 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 7435	. . . . . . 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 2795	. . . . . 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 50941	. . . . . . . . . . 11 ⊢ 1 ≠ 3
17		neeq1 3017	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 ≠ 3 ↔ 1 ≠ 3))
18	16, 17	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 ≠ 3)
19	18	neneqd 2960	. . . . . . . . 9 ⊢ (𝑥 = 1 → ¬ 𝑥 = 3)
20	19	iffalsed 4492	. . . . . . . 8 ⊢ (𝑥 = 1 → if(𝑥 = 3, ((𝑃‘3)↑2), 0) = 0)
21	20	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 = 1 → ((((𝑃‘1)↑2) + 0) + if(𝑥 = 3, ((𝑃‘3)↑2), 0)) = ((((𝑃‘1)↑2) + 0) + 0))
22	21	oveq1d 7435	. . . . . 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 11308	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
24		1lt4 12521	. . . . . . . . . . . . 13 ⊢ 1 < 4
25	23, 24	ltneii 11423	. . . . . . . . . . . 12 ⊢ 1 ≠ 4
26		neeq1 3017	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ≠ 4 ↔ 1 ≠ 4))
27	25, 26	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → 𝑥 ≠ 4)
28	27	neneqd 2960	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ¬ 𝑥 = 4)
29	28	iffalsed 4492	. . . . . . . . 9 ⊢ (𝑥 = 1 → if(𝑥 = 4, ((𝑃‘1) · (𝑃‘2)), 0) = 0)
30	29	oveq1d 7435	. . . . . . . 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 7435	. . . . . . 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 7436	. . . . . 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 2799	. . . . 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 12525	. . . . . . . . . . . 12 ⊢ 1 < 5
35	23, 34	ltneii 11423	. . . . . . . . . . 11 ⊢ 1 ≠ 5
36		neeq1 3017	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 ≠ 5 ↔ 1 ≠ 5))
37	35, 36	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 ≠ 5)
38	37	neneqd 2960	. . . . . . . . 9 ⊢ (𝑥 = 1 → ¬ 𝑥 = 5)
39	38	iffalsed 4492	. . . . . . . 8 ⊢ (𝑥 = 1 → if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0) = 0)
40	39	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 = 1 → (0 + if(𝑥 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) = (0 + 0))
41	40	oveq1d 7435	. . . . . 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 7436	. . . . 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 12530	. . . . . . . . . . 11 ⊢ 1 < 6
44	23, 43	ltneii 11423	. . . . . . . . . 10 ⊢ 1 ≠ 6
45		neeq1 3017	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 ≠ 6 ↔ 1 ≠ 6))
46	44, 45	mpbiri 261	. . . . . . . . 9 ⊢ (𝑥 = 1 → 𝑥 ≠ 6)
47	46	neneqd 2960	. . . . . . . 8 ⊢ (𝑥 = 1 → ¬ 𝑥 = 6)
48	47	iffalsed 4492	. . . . . . 7 ⊢ (𝑥 = 1 → if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0) = 0)
49	48	oveq2d 7436	. . . . . 6 ⊢ (𝑥 = 1 → ((0 + 0) + if(𝑥 = 6, ((𝑃‘3) · (𝑃‘1)), 0)) = ((0 + 0) + 0))
50	49	oveq2d 7436	. . . . 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 2799	. . . 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 50947	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃‘1) ∈ ℝ)
54	53	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 1) → (𝑃‘1) ∈ ℝ)
55	54	resqcld 14268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((𝑃‘1)↑2) ∈ ℝ)
56		0red 11311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 1) → 0 ∈ ℝ)
57	55, 56	readdcld 11338	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) ∈ ℝ)
58	57	recnd 11337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) ∈ ℂ)
59	58	addridd 11510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((((𝑃‘1)↑2) + 0) + 0) = (((𝑃‘1)↑2) + 0))
60	59	oveq1d 7435	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = ((((𝑃‘1)↑2) + 0) + ((0 + 0) + 0)))
61	55	recnd 11337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((𝑃‘1)↑2) ∈ ℂ)
62	61	addridd 11510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + 0) = ((𝑃‘1)↑2))
63	62	oveq1d 7435	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((((𝑃‘1)↑2) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + ((0 + 0) + 0)))
64	60, 63	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + ((0 + 0) + 0)))
65	56, 56	readdcld 11338	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) ∈ ℝ)
66	65	recnd 11337	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) ∈ ℂ)
67	66	addridd 11510	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → ((0 + 0) + 0) = (0 + 0))
68	67	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + (0 + 0)))
69		00id 11485	. . . . . 6 ⊢ (0 + 0) = 0
70	69	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 1) → (0 + 0) = 0)
71	70	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((𝑃‘1)↑2) + (0 + 0)) = (((𝑃‘1)↑2) + 0))
72	64, 68, 71	3eqtrd 2799	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 1) → (((((𝑃‘1)↑2) + 0) + 0) + ((0 + 0) + 0)) = (((𝑃‘1)↑2) + 0))
73	52, 72, 62	3eqtrd 2799	. 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 12727	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
75		6nn 12432	. . . . 5 ⊢ 6 ∈ ℕ
76	75	nnzi 12720	. . . 4 ⊢ 6 ∈ ℤ
77	76	a1i 11	. . 3 ⊢ (𝜑 → 6 ∈ ℤ)
78		1le1 11944	. . . 4 ⊢ 1 ≤ 1
79	78	a1i 11	. . 3 ⊢ (𝜑 → 1 ≤ 1)
80		6re 12433	. . . . 5 ⊢ 6 ∈ ℝ
81	23, 80, 43	ltleii 11433	. . . 4 ⊢ 1 ≤ 6
82	81	a1i 11	. . 3 ⊢ (𝜑 → 1 ≤ 6)
83	74, 77, 74, 79, 82	elfzd 13647	. 2 ⊢ (𝜑 → 1 ∈ (1...6))
84	53	resqcld 14268	. 2 ⊢ (𝜑 → ((𝑃‘1)↑2) ∈ ℝ)
85	2, 73, 83, 84	fvmptd 7001	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 1569 ∈ wcel 2144 ≠ wne 2955 ifcif 4481 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8847 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ≤ cle 11344 2c2 12397 3c3 12398 4c4 12399 5c5 12400 6c6 12401 ℤcz 12693 ...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-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 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-nel 3062 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-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 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-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 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-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 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-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-seq 14145 df-exp 14205 df-veronese 50969

This theorem is used by: veronesevrowd 50979 veroquadgsumlem 50983

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