veroquadgsumlem - Mathbox for Jiamin Zhao

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Theorem veroquadgsumlem 50983

Description: Lemma for veroquadmodzerod 50984. Express the common homogeneous quadratic equation in ℝ_fld Σ_g form using the Veronese matrix 𝑉. (Contributed by Jiamin Zhao, 19-Aug-2026.)

**Hypotheses**
Ref	Expression
veroquad.a	⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
veroquad.f	⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
veroquad.k	⊢ (𝜑 → 𝐾:(1...6)⟶ℝ)
veroquad.q	⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)

**Assertion**
Ref	Expression
veroquadgsumlem	⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝ_fld Σ_g (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))) = 0)

Distinct variable groups: 𝐴,𝑖,𝑗 𝑖,𝑉,𝑗 𝜑,𝑖 𝑗,𝐾 Allowed substitution hints: 𝜑(𝑗) 𝐾(𝑖)

Proof of Theorem veroquadgsumlem Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rebase 21912	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		replusg 21916	. . . 4 ⊢ + = (+_g‘ℝ_fld)
3		refld 21925	. . . . . 6 ⊢ ℝ_fld ∈ Field
4	3	elexi 3472	. . . . 5 ⊢ ℝ_fld ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ℝ_fld ∈ V)
6		6nn 12432	. . . . . 6 ⊢ 6 ∈ ℕ
7		nnuz 13004	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtri 2858	. . . . 5 ⊢ 6 ∈ (ℤ_≥‘1)
9	8	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 6 ∈ (ℤ_≥‘1))
10		veroquad.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾:(1...6)⟶ℝ)
11	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → 𝐾:(1...6)⟶ℝ)
12		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → 𝑣 ∈ (1...6))
13	11, 12	ffvelcdmd 7085	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → (𝐾‘𝑣) ∈ ℝ)
14		veroquad.a	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
15		veroquad.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
16	14, 15	veronesematrowd 50981	. . . . . . . . . 10 ⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))
17		fvexd 6900	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (veronese‘(𝐴‘𝑖)) ∈ V)
18	16, 17	fvmpt2d 7007	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (curry 𝑉‘𝑖) = (veronese‘(𝐴‘𝑖)))
19	18	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → (curry 𝑉‘𝑖) = (veronese‘(𝐴‘𝑖)))
20	19	fveq1d 6887	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘𝑣) = ((veronese‘(𝐴‘𝑖))‘𝑣))
21	15	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
22		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → 𝑖 ∈ (1...6))
23	21, 22	ffvelcdmd 7085	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → (𝐴‘𝑖) ∈ (ℝ ↑_m (1...3)))
24		veronesefvcl 50972	. . . . . . . 8 ⊢ (((𝐴‘𝑖) ∈ (ℝ ↑_m (1...3)) ∧ 𝑣 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘𝑣) ∈ ℝ)
25	23, 24	sylancom 600	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘𝑣) ∈ ℝ)
26	20, 25	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘𝑣) ∈ ℝ)
27	13, 26	remulcld 11339	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) ∈ ℝ)
28	27	fmpttd 7115	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))):(1...6)⟶ℝ)
29	1, 2, 5, 9, 28	gsumval2 18875	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝ_fld Σ_g (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘6))
30		5nn 12429	. . . . . . 7 ⊢ 5 ∈ ℕ
31	30, 7	eleqtri 2858	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘1)
32		seqp1 14159	. . . . . 6 ⊢ (5 ∈ (ℤ_≥‘1) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(5 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(5 + 1))))
33	31, 32	ax-mp 5	. . . . 5 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(5 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(5 + 1)))
34		5p1e6 12489	. . . . . 6 ⊢ (5 + 1) = 6
35	34	fveq2i 6888	. . . . 5 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(5 + 1)) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘6)
36	34	fveq2i 6888	. . . . . 6 ⊢ ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(5 + 1)) = ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6)
37	36	oveq2i 7431	. . . . 5 ⊢ ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(5 + 1))) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6))
38	33, 35, 37	3eqtr3i 2791	. . . 4 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘6) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6))
39		4nn 12426	. . . . . . . . 9 ⊢ 4 ∈ ℕ
40	39, 7	eleqtri 2858	. . . . . . . 8 ⊢ 4 ∈ (ℤ_≥‘1)
41		seqp1 14159	. . . . . . . 8 ⊢ (4 ∈ (ℤ_≥‘1) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(4 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(4 + 1))))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(4 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(4 + 1)))
43		4p1e5 12488	. . . . . . . 8 ⊢ (4 + 1) = 5
44	43	fveq2i 6888	. . . . . . 7 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(4 + 1)) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5)
45	43	fveq2i 6888	. . . . . . . 8 ⊢ ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(4 + 1)) = ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5)
46	45	oveq2i 7431	. . . . . . 7 ⊢ ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(4 + 1))) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5))
47	42, 44, 46	3eqtr3i 2791	. . . . . 6 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5))
48		3nn 12422	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
49	48, 7	eleqtri 2858	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
50		seqp1 14159	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(3 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(3 + 1))))
51	49, 50	ax-mp 5	. . . . . . . . 9 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(3 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(3 + 1)))
52		3p1e4 12487	. . . . . . . . . 10 ⊢ (3 + 1) = 4
53	52	fveq2i 6888	. . . . . . . . 9 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(3 + 1)) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4)
54	52	fveq2i 6888	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(3 + 1)) = ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4)
55	54	oveq2i 7431	. . . . . . . . 9 ⊢ ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(3 + 1))) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4))
56	51, 53, 55	3eqtr3i 2791	. . . . . . . 8 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4))
57		2eluzge1 13009	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
58		seqp1 14159	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(2 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(2 + 1))))
59	57, 58	ax-mp 5	. . . . . . . . . . 11 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(2 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(2 + 1)))
60		2p1e3 12484	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
61	60	fveq2i 6888	. . . . . . . . . . 11 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(2 + 1)) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3)
62	60	fveq2i 6888	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(2 + 1)) = ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3)
63	62	oveq2i 7431	. . . . . . . . . . 11 ⊢ ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(2 + 1))) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3))
64	59, 61, 63	3eqtr3i 2791	. . . . . . . . . 10 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3))
65		1nn 12346	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ
66	65, 7	eleqtri 2858	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (ℤ_≥‘1)
67		seqp1 14159	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (ℤ_≥‘1) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(1 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(1 + 1))))
68	66, 67	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(1 + 1)) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(1 + 1)))
69		1p1e2 12466	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
70	69	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘(1 + 1)) = (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2)
71	69	fveq2i 6888	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(1 + 1)) = ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2)
72	71	oveq2i 7431	. . . . . . . . . . . . 13 ⊢ ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘(1 + 1))) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2))
73	68, 70, 72	3eqtr3i 2791	. . . . . . . . . . . 12 ⊢ (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) = ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2))
74		1z 12726	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
75		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))) = (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))
76		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 1 → (𝐾‘𝑣) = (𝐾‘1))
77		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 1 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘1))
78	76, 77	oveq12d 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 1 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)))
79		1re 11308	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
80		6re 12433	. . . . . . . . . . . . . . . . . . 19 ⊢ 6 ∈ ℝ
81		1lt6 12530	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 6
82	79, 80, 81	ltleii 11433	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 6
83		elfz1b 13727	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
84	65, 6, 82, 83	mpbir3an 1360	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (1...6)
85	84	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 1 ∈ (1...6))
86		ovexd 7455	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)) ∈ V)
87	75, 78, 85, 86	fvmptd3 7017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘1) = ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)))
88	18	fveq1d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘1) = ((veronese‘(𝐴‘𝑖))‘1))
89	15	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐴‘𝑖) ∈ (ℝ ↑_m (1...3)))
90	89	veronesev1lem 50973	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘1) = (((𝐴‘𝑖)‘1)↑2))
91	88, 90	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘1) = (((𝐴‘𝑖)‘1)↑2))
92	91	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)) = ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)))
93	87, 92	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘1) = ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)))
94	74, 93	seq1i 14158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) = ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)))
95		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 2 → (𝐾‘𝑣) = (𝐾‘2))
96		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 2 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘2))
97	95, 96	oveq12d 7438	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 2 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)))
98		2nn 12416	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
99		2re 12417	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
100		2lt6 12529	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 6
101	99, 80, 100	ltleii 11433	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 6
102		elfz1b 13727	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ (1...6) ↔ (2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 2 ≤ 6))
103	98, 6, 101, 102	mpbir3an 1360	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (1...6)
104	103	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 2 ∈ (1...6))
105		ovexd 7455	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)) ∈ V)
106	75, 97, 104, 105	fvmptd3 7017	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2) = ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)))
107	18	fveq1d 6887	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘2) = ((veronese‘(𝐴‘𝑖))‘2))
108	89	veronesev2lem 50974	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘2) = (((𝐴‘𝑖)‘2)↑2))
109	107, 108	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘2) = (((𝐴‘𝑖)‘2)↑2))
110	109	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)) = ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)))
111	106, 110	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2) = ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)))
112	94, 111	oveq12d 7438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2)) = (((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))))
113	73, 112	eqtrid 2807	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) = (((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))))
114		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 3 → (𝐾‘𝑣) = (𝐾‘3))
115		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 3 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘3))
116	114, 115	oveq12d 7438	. . . . . . . . . . . . 13 ⊢ (𝑣 = 3 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)))
117		3re 12423	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
118		3lt6 12528	. . . . . . . . . . . . . . . 16 ⊢ 3 < 6
119	117, 80, 118	ltleii 11433	. . . . . . . . . . . . . . 15 ⊢ 3 ≤ 6
120		elfz1b 13727	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ (1...6) ↔ (3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 3 ≤ 6))
121	48, 6, 119, 120	mpbir3an 1360	. . . . . . . . . . . . . 14 ⊢ 3 ∈ (1...6)
122	121	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 3 ∈ (1...6))
123		ovexd 7455	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)) ∈ V)
124	75, 116, 122, 123	fvmptd3 7017	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3) = ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)))
125	18	fveq1d 6887	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘3) = ((veronese‘(𝐴‘𝑖))‘3))
126	89	veronesev3lem 50975	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘3) = (((𝐴‘𝑖)‘3)↑2))
127	125, 126	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘3) = (((𝐴‘𝑖)‘3)↑2))
128	127	oveq2d 7436	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)) = ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)))
129	124, 128	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3) = ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)))
130	113, 129	oveq12d 7438	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘2) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3)) = ((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))))
131	64, 130	eqtrid 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) = ((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))))
132		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑣 = 4 → (𝐾‘𝑣) = (𝐾‘4))
133		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑣 = 4 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘4))
134	132, 133	oveq12d 7438	. . . . . . . . . . 11 ⊢ (𝑣 = 4 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)))
135		4re 12427	. . . . . . . . . . . . . 14 ⊢ 4 ∈ ℝ
136		4lt6 12527	. . . . . . . . . . . . . 14 ⊢ 4 < 6
137	135, 80, 136	ltleii 11433	. . . . . . . . . . . . 13 ⊢ 4 ≤ 6
138		elfz1b 13727	. . . . . . . . . . . . 13 ⊢ (4 ∈ (1...6) ↔ (4 ∈ ℕ ∧ 6 ∈ ℕ ∧ 4 ≤ 6))
139	39, 6, 137, 138	mpbir3an 1360	. . . . . . . . . . . 12 ⊢ 4 ∈ (1...6)
140	139	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 4 ∈ (1...6))
141		ovexd 7455	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)) ∈ V)
142	75, 134, 140, 141	fvmptd3 7017	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4) = ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)))
143	18	fveq1d 6887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘4) = ((veronese‘(𝐴‘𝑖))‘4))
144	89	veronesev4lem 50976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘4) = (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))
145	143, 144	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘4) = (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))
146	145	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)) = ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))))
147	142, 146	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4) = ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))))
148	131, 147	oveq12d 7438	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘3) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4)) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))))
149	56, 148	eqtrid 2807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))))
150		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑣 = 5 → (𝐾‘𝑣) = (𝐾‘5))
151		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑣 = 5 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘5))
152	150, 151	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑣 = 5 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)))
153		5re 12430	. . . . . . . . . . . 12 ⊢ 5 ∈ ℝ
154		5lt6 12526	. . . . . . . . . . . 12 ⊢ 5 < 6
155	153, 80, 154	ltleii 11433	. . . . . . . . . . 11 ⊢ 5 ≤ 6
156		elfz1b 13727	. . . . . . . . . . 11 ⊢ (5 ∈ (1...6) ↔ (5 ∈ ℕ ∧ 6 ∈ ℕ ∧ 5 ≤ 6))
157	30, 6, 155, 156	mpbir3an 1360	. . . . . . . . . 10 ⊢ 5 ∈ (1...6)
158	157	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 5 ∈ (1...6))
159		ovexd 7455	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)) ∈ V)
160	75, 152, 158, 159	fvmptd3 7017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5) = ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)))
161	18	fveq1d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘5) = ((veronese‘(𝐴‘𝑖))‘5))
162	89	veronesev5lem 50977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘5) = (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))
163	161, 162	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘5) = (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))
164	163	oveq2d 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)) = ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))))
165	160, 164	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5) = ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))))
166	149, 165	oveq12d 7438	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘4) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5)) = ((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))))
167	47, 166	eqtrid 2807	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) = ((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))))
168		fveq2 6885	. . . . . . . 8 ⊢ (𝑣 = 6 → (𝐾‘𝑣) = (𝐾‘6))
169		fveq2 6885	. . . . . . . 8 ⊢ (𝑣 = 6 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘6))
170	168, 169	oveq12d 7438	. . . . . . 7 ⊢ (𝑣 = 6 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)))
171	80	leidi 11850	. . . . . . . . 9 ⊢ 6 ≤ 6
172		elfz1b 13727	. . . . . . . . 9 ⊢ (6 ∈ (1...6) ↔ (6 ∈ ℕ ∧ 6 ∈ ℕ ∧ 6 ≤ 6))
173	6, 6, 171, 172	mpbir3an 1360	. . . . . . . 8 ⊢ 6 ∈ (1...6)
174	173	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 6 ∈ (1...6))
175		ovexd 7455	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)) ∈ V)
176	75, 170, 174, 175	fvmptd3 7017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6) = ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)))
177	18	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘6) = ((veronese‘(𝐴‘𝑖))‘6))
178	89	veronesev6lem 50978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((veronese‘(𝐴‘𝑖))‘6) = (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))
179	177, 178	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘6) = (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))
180	179	oveq2d 7436	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)) = ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))
181	176, 180	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6) = ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))
182	167, 181	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘5) + ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6)) = (((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))))
183	38, 182	eqtrid 2807	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘6) = (((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))))
184	10	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → 𝐾:(1...6)⟶ℝ)
185	184, 85	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘1) ∈ ℝ)
186	89	rr3fv1cld 50947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘1) ∈ ℝ)
187	186	resqcld 14268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘1)↑2) ∈ ℝ)
188	185, 187	remulcld 11339	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) ∈ ℝ)
189	188	recnd 11337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) ∈ ℂ)
190	184, 104	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘2) ∈ ℝ)
191	89	rr3fv2cld 50948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘2) ∈ ℝ)
192	191	resqcld 14268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘2)↑2) ∈ ℝ)
193	190, 192	remulcld 11339	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) ∈ ℝ)
194	193	recnd 11337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) ∈ ℂ)
195	189, 194	addcld 11328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) ∈ ℂ)
196	184, 122	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘3) ∈ ℝ)
197	89	rr3fv3cld 50949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘3) ∈ ℝ)
198	197	resqcld 14268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘3)↑2) ∈ ℝ)
199	196, 198	remulcld 11339	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) ∈ ℝ)
200	199	recnd 11337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) ∈ ℂ)
201	195, 200	addcld 11328	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) ∈ ℂ)
202	184, 140	ffvelcdmd 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘4) ∈ ℝ)
203	186, 191	remulcld 11339	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)) ∈ ℝ)
204	202, 203	remulcld 11339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) ∈ ℝ)
205	204	recnd 11337	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) ∈ ℂ)
206	184, 158	ffvelcdmd 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘5) ∈ ℝ)
207	191, 197	remulcld 11339	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)) ∈ ℝ)
208	206, 207	remulcld 11339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) ∈ ℝ)
209	208	recnd 11337	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) ∈ ℂ)
210	201, 205, 209	addassd 11331	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + (((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))))))
211	210	oveq1d 7435	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))) = ((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + (((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))))
212	205, 209	addcld 11328	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) ∈ ℂ)
213	184, 174	ffvelcdmd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘6) ∈ ℝ)
214	197, 186	remulcld 11339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)) ∈ ℝ)
215	213, 214	remulcld 11339	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) ∈ ℝ)
216	215	recnd 11337	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) ∈ ℂ)
217	201, 212, 216	addassd 11331	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + (((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))))
218	211, 217	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)))) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))))
219	29, 183, 218	3eqtrd 2799	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝ_fld Σ_g (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))) = (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))))
220		fveq2 6885	. . . . . 6 ⊢ (𝑣 = 𝑗 → (𝐾‘𝑣) = (𝐾‘𝑗))
221		fveq2 6885	. . . . . 6 ⊢ (𝑣 = 𝑗 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘𝑗))
222	220, 221	oveq12d 7438	. . . . 5 ⊢ (𝑣 = 𝑗 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))
223	222	cbvmptv 5208	. . . 4 ⊢ (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))) = (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))
224	223	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))) = (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗))))
225	224	oveq2d 7436	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝ_fld Σ_g (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))) = (ℝ_fld Σ_g (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))))
226		veroquad.q	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)
227	219, 225, 226	3eqtr3d 2803	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝ_fld Σ_g (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 class class class wbr 5102 ↦ cmpt 5185 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 curry ccur 8282 ↑_m cmap 8847 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ≤ cle 11344 ℕcn 12335 2c2 12397 3c3 12398 4c4 12399 5c5 12400 6c6 12401 ℤ_≥cuz 12965 ...cfz 13639 seqcseq 14144 ↑cexp 14204 Σ_g cgsu 17611 Fieldcfield 20981 ℝ_fldcrefld 21910 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 ax-addf 11279

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-rmo 3365 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-tp 4588 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-tpos 8243 df-cur 8284 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-seq 14145 df-exp 14205 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-0g 17612 df-gsum 17613 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-grp 19147 df-minusg 19148 df-subg 19333 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-subrng 20798 df-subrg 20822 df-drng 20982 df-field 20983 df-cnfld 21679 df-refld 21911 df-veronese 50969

This theorem is used by: veroquadmodzerod 50984

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