veroquadgsumlem - Mathbox for Jiamin Zhao

	Mathbox for Jiamin Zhao		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > veroquadgsumlem		Structured version Visualization version GIF version

Theorem veroquadgsumlem 50816

Description: Lemma for veroquadmodzerod 50817. 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 21819	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		replusg 21823	. . . 4 ⊢ + = (+_g‘ℝ_fld)
3		refld 21832	. . . . . 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 12354	. . . . . 6 ⊢ 6 ∈ ℕ
7		nnuz 12926	. . . . . 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 7078	. . . . . 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 50814	. . . . . . . . . 10 ⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))
17		fvexd 6893	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (veronese‘(𝐴‘𝑖)) ∈ V)
18	16, 17	fvmpt2d 7000	. . . . . . . . 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 6880	. . . . . . 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 7078	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → (𝐴‘𝑖) ∈ (ℝ ↑_m (1...3)))
24		veronesefvcl 50805	. . . . . . . 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 11263	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...6)) ∧ 𝑣 ∈ (1...6)) → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) ∈ ℝ)
28	27	fmpttd 7108	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))):(1...6)⟶ℝ)
29	1, 2, 5, 9, 28	gsumval2 18788	. . 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 12351	. . . . . . 7 ⊢ 5 ∈ ℕ
31	30, 7	eleqtri 2858	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘1)
32		seqp1 14080	. . . . . 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 12411	. . . . . 6 ⊢ (5 + 1) = 6
35	34	fveq2i 6881	. . . . 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 6881	. . . . . 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 7424	. . . . 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 12348	. . . . . . . . 9 ⊢ 4 ∈ ℕ
40	39, 7	eleqtri 2858	. . . . . . . 8 ⊢ 4 ∈ (ℤ_≥‘1)
41		seqp1 14080	. . . . . . . 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 12410	. . . . . . . 8 ⊢ (4 + 1) = 5
44	43	fveq2i 6881	. . . . . . 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 6881	. . . . . . . 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 7424	. . . . . . 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 12344	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
49	48, 7	eleqtri 2858	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
50		seqp1 14080	. . . . . . . . . 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 12409	. . . . . . . . . 10 ⊢ (3 + 1) = 4
53	52	fveq2i 6881	. . . . . . . . 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 6881	. . . . . . . . . 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 7424	. . . . . . . . 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 12931	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
58		seqp1 14080	. . . . . . . . . . . 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 12406	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
61	60	fveq2i 6881	. . . . . . . . . . 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 6881	. . . . . . . . . . . 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 7424	. . . . . . . . . . 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 12268	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ
66	65, 7	eleqtri 2858	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (ℤ_≥‘1)
67		seqp1 14080	. . . . . . . . . . . . . 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 12388	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
70	69	fveq2i 6881	. . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . 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 12648	. . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 1 → (𝐾‘𝑣) = (𝐾‘1))
77		fveq2 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 1 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘1))
78	76, 77	oveq12d 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 1 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)))
79		1re 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
80		6re 12355	. . . . . . . . . . . . . . . . . . 19 ⊢ 6 ∈ ℝ
81		1lt6 12452	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 6
82	79, 80, 81	ltleii 11357	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 6
83		elfz1b 13648	. . . . . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)) ∈ V)
87	75, 78, 85, 86	fvmptd3 7010	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘1) = ((𝐾‘1) · ((curry 𝑉‘𝑖)‘1)))
88	18	fveq1d 6880	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘1) = ((veronese‘(𝐴‘𝑖))‘1))
89	15	ffvelcdmda 7077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐴‘𝑖) ∈ (ℝ ↑_m (1...3)))
90	89	veronesev1lem 50806	. . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . 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 14079	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (seq1( + , (𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣))))‘1) = ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)))
95		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 2 → (𝐾‘𝑣) = (𝐾‘2))
96		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 2 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘2))
97	95, 96	oveq12d 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 2 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)))
98		2nn 12338	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
99		2re 12339	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
100		2lt6 12451	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 6
101	99, 80, 100	ltleii 11357	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 6
102		elfz1b 13648	. . . . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)) ∈ V)
106	75, 97, 104, 105	fvmptd3 7010	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘2) = ((𝐾‘2) · ((curry 𝑉‘𝑖)‘2)))
107	18	fveq1d 6880	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘2) = ((veronese‘(𝐴‘𝑖))‘2))
108	89	veronesev2lem 50807	. . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . 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 7431	. . . . . . . . . . . 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 6878	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 3 → (𝐾‘𝑣) = (𝐾‘3))
115		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 3 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘3))
116	114, 115	oveq12d 7431	. . . . . . . . . . . . 13 ⊢ (𝑣 = 3 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)))
117		3re 12345	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
118		3lt6 12450	. . . . . . . . . . . . . . . 16 ⊢ 3 < 6
119	117, 80, 118	ltleii 11357	. . . . . . . . . . . . . . 15 ⊢ 3 ≤ 6
120		elfz1b 13648	. . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)) ∈ V)
124	75, 116, 122, 123	fvmptd3 7010	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘3) = ((𝐾‘3) · ((curry 𝑉‘𝑖)‘3)))
125	18	fveq1d 6880	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘3) = ((veronese‘(𝐴‘𝑖))‘3))
126	89	veronesev3lem 50808	. . . . . . . . . . . . . 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 7429	. . . . . . . . . . . 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 7431	. . . . . . . . . 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 6878	. . . . . . . . . . . 12 ⊢ (𝑣 = 4 → (𝐾‘𝑣) = (𝐾‘4))
133		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑣 = 4 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘4))
134	132, 133	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑣 = 4 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)))
135		4re 12349	. . . . . . . . . . . . . 14 ⊢ 4 ∈ ℝ
136		4lt6 12449	. . . . . . . . . . . . . 14 ⊢ 4 < 6
137	135, 80, 136	ltleii 11357	. . . . . . . . . . . . 13 ⊢ 4 ≤ 6
138		elfz1b 13648	. . . . . . . . . . . . 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 7448	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)) ∈ V)
142	75, 134, 140, 141	fvmptd3 7010	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘4) = ((𝐾‘4) · ((curry 𝑉‘𝑖)‘4)))
143	18	fveq1d 6880	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘4) = ((veronese‘(𝐴‘𝑖))‘4))
144	89	veronesev4lem 50809	. . . . . . . . . . . 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 7429	. . . . . . . . . 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 7431	. . . . . . . 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 6878	. . . . . . . . . 10 ⊢ (𝑣 = 5 → (𝐾‘𝑣) = (𝐾‘5))
151		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑣 = 5 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘5))
152	150, 151	oveq12d 7431	. . . . . . . . 9 ⊢ (𝑣 = 5 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)))
153		5re 12352	. . . . . . . . . . . 12 ⊢ 5 ∈ ℝ
154		5lt6 12448	. . . . . . . . . . . 12 ⊢ 5 < 6
155	153, 80, 154	ltleii 11357	. . . . . . . . . . 11 ⊢ 5 ≤ 6
156		elfz1b 13648	. . . . . . . . . . 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 7448	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)) ∈ V)
160	75, 152, 158, 159	fvmptd3 7010	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘5) = ((𝐾‘5) · ((curry 𝑉‘𝑖)‘5)))
161	18	fveq1d 6880	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘5) = ((veronese‘(𝐴‘𝑖))‘5))
162	89	veronesev5lem 50810	. . . . . . . . . 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 7429	. . . . . . . 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 7431	. . . . . 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 6878	. . . . . . . 8 ⊢ (𝑣 = 6 → (𝐾‘𝑣) = (𝐾‘6))
169		fveq2 6878	. . . . . . . 8 ⊢ (𝑣 = 6 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘6))
170	168, 169	oveq12d 7431	. . . . . . 7 ⊢ (𝑣 = 6 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)))
171	80	leidi 11772	. . . . . . . . 9 ⊢ 6 ≤ 6
172		elfz1b 13648	. . . . . . . . 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 7448	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)) ∈ V)
176	75, 170, 174, 175	fvmptd3 7010	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝑣 ∈ (1...6) ↦ ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)))‘6) = ((𝐾‘6) · ((curry 𝑉‘𝑖)‘6)))
177	18	fveq1d 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((curry 𝑉‘𝑖)‘6) = ((veronese‘(𝐴‘𝑖))‘6))
178	89	veronesev6lem 50811	. . . . . . . 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 7429	. . . . . 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 7431	. . . 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 7078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘1) ∈ ℝ)
186	89	rr3fv1cld 50780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘1) ∈ ℝ)
187	186	resqcld 14189	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘1)↑2) ∈ ℝ)
188	185, 187	remulcld 11263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) ∈ ℝ)
189	188	recnd 11261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) ∈ ℂ)
190	184, 104	ffvelcdmd 7078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘2) ∈ ℝ)
191	89	rr3fv2cld 50781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘2) ∈ ℝ)
192	191	resqcld 14189	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘2)↑2) ∈ ℝ)
193	190, 192	remulcld 11263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) ∈ ℝ)
194	193	recnd 11261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) ∈ ℂ)
195	189, 194	addcld 11252	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) ∈ ℂ)
196	184, 122	ffvelcdmd 7078	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘3) ∈ ℝ)
197	89	rr3fv3cld 50782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐴‘𝑖)‘3) ∈ ℝ)
198	197	resqcld 14189	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘3)↑2) ∈ ℝ)
199	196, 198	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) ∈ ℝ)
200	199	recnd 11261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) ∈ ℂ)
201	195, 200	addcld 11252	. . . . . 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 7078	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘4) ∈ ℝ)
203	186, 191	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)) ∈ ℝ)
204	202, 203	remulcld 11263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) ∈ ℝ)
205	204	recnd 11261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) ∈ ℂ)
206	184, 158	ffvelcdmd 7078	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘5) ∈ ℝ)
207	191, 197	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)) ∈ ℝ)
208	206, 207	remulcld 11263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) ∈ ℝ)
209	208	recnd 11261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) ∈ ℂ)
210	201, 205, 209	addassd 11255	. . . . 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 7428	. . . 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 11252	. . . . 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 7078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (𝐾‘6) ∈ ℝ)
214	197, 186	remulcld 11263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)) ∈ ℝ)
215	213, 214	remulcld 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) ∈ ℝ)
216	215	recnd 11261	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) ∈ ℂ)
217	201, 212, 216	addassd 11255	. . . 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 6878	. . . . . 6 ⊢ (𝑣 = 𝑗 → (𝐾‘𝑣) = (𝐾‘𝑗))
221		fveq2 6878	. . . . . 6 ⊢ (𝑣 = 𝑗 → ((curry 𝑉‘𝑖)‘𝑣) = ((curry 𝑉‘𝑖)‘𝑗))
222	220, 221	oveq12d 7431	. . . . 5 ⊢ (𝑣 = 𝑗 → ((𝐾‘𝑣) · ((curry 𝑉‘𝑖)‘𝑣)) = ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))
223	222	cbvmptv 5209	. . . 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 7429	. 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 1570 ∈ wcel 2145 Vcvv 3450 class class class wbr 5103 ↦ cmpt 5186 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 curry ccur 8263 ↑_m cmap 8826 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 ≤ cle 11268 ℕcn 12257 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 ℤ_≥cuz 12887 ...cfz 13561 seqcseq 14065 ↑cexp 14125 Σ_g cgsu 17525 Fieldcfield 20891 ℝ_fldcrefld 21817 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-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-addf 11203

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 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 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-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 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-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-cur 8265 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-seq 14066 df-exp 14126 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-0g 17526 df-gsum 17527 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-grp 19060 df-minusg 19061 df-subg 19246 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-cring 20375 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-subrng 20708 df-subrg 20732 df-drng 20892 df-field 20893 df-cnfld 21586 df-refld 21818 df-veronese 50802

This theorem is used by: veroquadmodzerod 50817

W3C validator