veroquadmodzerod - Mathbox for Jiamin Zhao

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Theorem veroquadmodzerod 50799

Description: The columns of the Veronese matrix, weighted by the coefficients 𝐾, sum to the zero vector of ℝ_fld freeLMod (1...6). (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
veroquadmodzerod	⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = (0_g‘(ℝ_fld freeLMod (1...6))))

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

Proof of Theorem veroquadmodzerod Dummy variables 𝑛 𝑚 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		veroquad.k	. . . . . . 7 ⊢ (𝜑 → 𝐾:(1...6)⟶ℝ)
2	1	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝐾 Fn (1...6))
3		veroquad.a	. . . . . . . . . . . 12 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
4		2fveq3 6887	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
5	4	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
6		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
7	5, 6	cbvmpov 7511	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗)) = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
8	3, 7	eqtri 2785	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
9	8	tposmpo 8264	. . . . . . . . . 10 ⊢ tpos 𝑉 = (𝑣 ∈ (1...6), 𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → tpos 𝑉 = (𝑣 ∈ (1...6), 𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)))
11		veroquad.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
12	11	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
13		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝑢 ∈ (1...6))
14	12, 13	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → (𝐴‘𝑢) ∈ (ℝ ↑_m (1...3)))
15		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝑣 ∈ (1...6))
16		veronesefvcl 50787	. . . . . . . . . 10 ⊢ (((𝐴‘𝑢) ∈ (ℝ ↑_m (1...3)) ∧ 𝑣 ∈ (1...6)) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
17	14, 15, 16	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
18	10, 17	fmpod 8073	. . . . . . . 8 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
19		ovex 7449	. . . . . . . . . 10 ⊢ (1...6) ∈ V
20		1nn 12269	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
21		6nn 12355	. . . . . . . . . . . 12 ⊢ 6 ∈ ℕ
22		1re 11233	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
23		6re 12356	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℝ
24		1lt6 12453	. . . . . . . . . . . . 13 ⊢ 1 < 6
25	22, 23, 24	ltleii 11358	. . . . . . . . . . . 12 ⊢ 1 ≤ 6
26		elfz1b 13648	. . . . . . . . . . . 12 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
27	20, 21, 25, 26	mpbir3an 1360	. . . . . . . . . . 11 ⊢ 1 ∈ (1...6)
28	27	ne0ii 4293	. . . . . . . . . 10 ⊢ (1...6) ≠ ∅
29		eldifsn 4751	. . . . . . . . . 10 ⊢ ((1...6) ∈ (V ∖ {∅}) ↔ ((1...6) ∈ V ∧ (1...6) ≠ ∅))
30	19, 28, 29	mpbir2an 724	. . . . . . . . 9 ⊢ (1...6) ∈ (V ∖ {∅})
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (1...6) ∈ (V ∖ {∅}))
32		reex 11216	. . . . . . . . 9 ⊢ ℝ ∈ V
33	32	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
34		curf 8872	. . . . . . . 8 ⊢ ((tpos 𝑉:((1...6) × (1...6))⟶ℝ ∧ (1...6) ∈ (V ∖ {∅}) ∧ ℝ ∈ V) → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
35	18, 31, 33, 34	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
36	35	ffnd 6707	. . . . . 6 ⊢ (𝜑 → curry tpos 𝑉 Fn (1...6))
37	19	a1i 11	. . . . . 6 ⊢ (𝜑 → (1...6) ∈ V)
38		inidm 4175	. . . . . 6 ⊢ ((1...6) ∩ (1...6)) = (1...6)
39		eqidd 2763	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) = (𝐾‘𝑛))
40	17	ralrimivva 3207	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ (1...6)∀𝑢 ∈ (1...6)((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
41	40	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → ∀𝑣 ∈ (1...6)∀𝑢 ∈ (1...6)((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
42	28	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (1...6) ≠ ∅)
43	19	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (1...6) ∈ V)
44		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → 𝑛 ∈ (1...6))
45	9, 41, 42, 43, 44	mpocurryvald 8271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)))
46		csbfv 6929	. . . . . . . . 9 ⊢ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣) = ((veronese‘(𝐴‘𝑢))‘𝑛)
47	46	mpteq2i 5205	. . . . . . . 8 ⊢ (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛))
48	45, 47	eqtrdi 2813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)))
49		2fveq3 6887	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
50	49	fveq1d 6884	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → ((veronese‘(𝐴‘𝑢))‘𝑛) = ((veronese‘(𝐴‘𝑖))‘𝑛))
51	50	cbvmptv 5213	. . . . . . 7 ⊢ (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
52	48, 51	eqtrdi 2813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)))
53	2, 36, 37, 37, 38, 39, 52	offval 7690	. . . . 5 ⊢ (𝜑 → (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉) = (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛)( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))(𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)))))
54		eqid 2762	. . . . . . 7 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
55		eqid 2762	. . . . . . 7 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
56		rebase 21818	. . . . . . 7 ⊢ ℝ = (Base‘ℝ_fld)
57	1	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ ℝ)
58		refld 21831	. . . . . . . . . . . . 13 ⊢ ℝ_fld ∈ Field
59	58	elexi 3475	. . . . . . . . . . . 12 ⊢ ℝ_fld ∈ V
60		fzfi 14036	. . . . . . . . . . . 12 ⊢ (1...6) ∈ Fin
61	54, 56	frlmfibas 21974	. . . . . . . . . . . 12 ⊢ ((ℝ_fld ∈ V ∧ (1...6) ∈ Fin) → (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6))))
62	59, 60, 61	mp2an 705	. . . . . . . . . . 11 ⊢ (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6)))
63	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6))))
64	63, 35	feq3dd 6693	. . . . . . . . 9 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(Base‘(ℝ_fld freeLMod (1...6))))
65	64	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
66	52, 65	eqeltrrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
67		eqid 2762	. . . . . . 7 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
68		remulr 21823	. . . . . . 7 ⊢ · = (._r‘ℝ_fld)
69	54, 55, 56, 43, 57, 66, 67, 68	frlmvscafval 21978	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → ((𝐾‘𝑛)( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))(𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))) = (((1...6) × {(𝐾‘𝑛)}) ∘_f · (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))))
70	69	mpteq2dva 5202	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛)( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))(𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)))) = (𝑛 ∈ (1...6) ↦ (((1...6) × {(𝐾‘𝑛)}) ∘_f · (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)))))
71		fvexd 6897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ V)
72		fnconstg 6767	. . . . . . . 8 ⊢ ((𝐾‘𝑛) ∈ V → ((1...6) × {(𝐾‘𝑛)}) Fn (1...6))
73	71, 72	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → ((1...6) × {(𝐾‘𝑛)}) Fn (1...6))
74		fvex 6895	. . . . . . . . 9 ⊢ ((veronese‘(𝐴‘𝑖))‘𝑛) ∈ V
75		eqid 2762	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
76	74, 75	fnmpti 6679	. . . . . . . 8 ⊢ (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) Fn (1...6)
77	76	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) Fn (1...6))
78		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → 𝑚 ∈ (1...6))
79		fvex 6895	. . . . . . . . 9 ⊢ (𝐾‘𝑛) ∈ V
80	79	fvconst2 7206	. . . . . . . 8 ⊢ (𝑚 ∈ (1...6) → (((1...6) × {(𝐾‘𝑛)})‘𝑚) = (𝐾‘𝑛))
81	78, 80	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (((1...6) × {(𝐾‘𝑛)})‘𝑚) = (𝐾‘𝑛))
82		2fveq3 6887	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑚)))
83	82	fveq1d 6884	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
84		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → 𝑚 ∈ (1...6))
85		fvexd 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) ∈ V)
86	75, 83, 84, 85	fvmptd3 7014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → ((𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))‘𝑚) = ((veronese‘(𝐴‘𝑚))‘𝑛))
87	86	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))‘𝑚) = ((veronese‘(𝐴‘𝑚))‘𝑛))
88	73, 77, 43, 43, 38, 81, 87	offval 7690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (((1...6) × {(𝐾‘𝑛)}) ∘_f · (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))) = (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))
89	88	mpteq2dva 5202	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (1...6) ↦ (((1...6) × {(𝐾‘𝑛)}) ∘_f · (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)))) = (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))))
90	53, 70, 89	3eqtrd 2801	. . . 4 ⊢ (𝜑 → (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉) = (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))))
91	90	oveq2d 7432	. . 3 ⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = ((ℝ_fld freeLMod (1...6)) Σ_g (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))))
92		eqid 2762	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
93		isfld 20902	. . . . . . . 8 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
94	58, 93	mpbi 233	. . . . . . 7 ⊢ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing)
95	94	simpli 489	. . . . . 6 ⊢ ℝ_fld ∈ DivRing
96		drngring 20896	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
97	95, 96	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
98	97	a1i 11	. . . 4 ⊢ (𝜑 → ℝ_fld ∈ Ring)
99	57	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (𝐾‘𝑛) ∈ ℝ)
100		simpll 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → 𝜑)
101	100, 11	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
102	101, 78	ffvelcdmd 7081	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (𝐴‘𝑚) ∈ (ℝ ↑_m (1...3)))
103		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → 𝑛 ∈ (1...6))
104		veronesefvcl 50787	. . . . . . . 8 ⊢ (((𝐴‘𝑚) ∈ (ℝ ↑_m (1...3)) ∧ 𝑛 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) ∈ ℝ)
105	102, 103, 104	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) ∈ ℝ)
106	99, 105	remulcld 11264	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)) ∈ ℝ)
107	106	fmpttd 7111	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))):(1...6)⟶ℝ)
108	32, 19	elmap 8881	. . . . 5 ⊢ ((𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))) ∈ (ℝ ↑_m (1...6)) ↔ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))):(1...6)⟶ℝ)
109	107, 108	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))) ∈ (ℝ ↑_m (1...6)))
110		eqid 2762	. . . . 5 ⊢ (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) = (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))
111	60	a1i 11	. . . . 5 ⊢ (𝜑 → (1...6) ∈ Fin)
112		fvexd 6897	. . . . 5 ⊢ (𝜑 → (0_g‘(ℝ_fld freeLMod (1...6))) ∈ V)
113	110, 111, 109, 112	fsuppmptdm 9349	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) finSupp (0_g‘(ℝ_fld freeLMod (1...6))))
114	54, 62, 92, 37, 37, 98, 109, 113	frlmgsum 21984	. . 3 ⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝑛 ∈ (1...6) ↦ (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))) = (𝑚 ∈ (1...6) ↦ (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))))
115	3, 11	veronesematrowd 50796	. . . . . . . . . . . . 13 ⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))
116	100, 115	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))
117		fvexd 6897	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (veronese‘(𝐴‘𝑚)) ∈ V)
118	82, 116, 78, 117	fvmptd4 7015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (curry 𝑉‘𝑚) = (veronese‘(𝐴‘𝑚)))
119	118	fveq1d 6884	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((curry 𝑉‘𝑚)‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
120	119	eqcomd 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) = ((curry 𝑉‘𝑚)‘𝑛))
121	120	oveq2d 7432	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)) = ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))
122	121	an32s 665	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (1...6)) ∧ 𝑛 ∈ (1...6)) → ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)) = ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))
123	122	mpteq2dva 5202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))) = (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛))))
124	123	oveq2d 7432	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) = (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))))
125	82	fveq1d 6884	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑗))
126		fveq2 6882	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → ((veronese‘(𝐴‘𝑚))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑛))
127	125, 126	cbvmpov 7511	. . . . . . 7 ⊢ (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗)) = (𝑚 ∈ (1...6), 𝑛 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑚))‘𝑛))
128	3, 127	eqtri 2785	. . . . . 6 ⊢ 𝑉 = (𝑚 ∈ (1...6), 𝑛 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑚))‘𝑛))
129		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (𝐴‘𝑖) = (𝐴‘𝑚))
130	129	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘1) = ((𝐴‘𝑚)‘1))
131	130	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1)↑2) = (((𝐴‘𝑚)‘1)↑2))
132	131	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) = ((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)))
133	129	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘2) = ((𝐴‘𝑚)‘2))
134	133	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2)↑2) = (((𝐴‘𝑚)‘2)↑2))
135	134	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) = ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2)))
136	132, 135	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) = (((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2))))
137	129	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘3) = ((𝐴‘𝑚)‘3))
138	137	oveq1d 7431	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3)↑2) = (((𝐴‘𝑚)‘3)↑2))
139	138	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) = ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2)))
140	136, 139	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) = ((((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2))))
141	130, 133	oveq12d 7434	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)) = (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2)))
142	141	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) = ((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))))
143	133, 137	oveq12d 7434	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)) = (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))
144	143	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) = ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3))))
145	142, 144	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) = (((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))) + ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))))
146	137, 130	oveq12d 7434	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)) = (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1)))
147	146	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) = ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))
148	145, 147	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((((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‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))) + ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1)))))
149	140, 148	oveq12d 7434	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → (((((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‘𝑚)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))) + ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))))
150	149	eqeq1d 2764	. . . . . . 7 ⊢ (𝑖 = 𝑚 → ((((((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 ↔ (((((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))) + ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))) = 0))
151		veroquad.q	. . . . . . . . 9 ⊢ ((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)
152	151	ralrimiva 3156	. . . . . . . 8 ⊢ (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)
153	152	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → ∀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)
154	150, 153, 84	rspcdva 3580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))) + ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))) = 0)
155	128, 11, 1, 154	veroquadgsumlem 50798	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))) = 0)
156	124, 155	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) = 0)
157	156	mpteq2dva 5202	. . 3 ⊢ (𝜑 → (𝑚 ∈ (1...6) ↦ (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))))) = (𝑚 ∈ (1...6) ↦ 0))
158	91, 114, 157	3eqtrd 2801	. 2 ⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = (𝑚 ∈ (1...6) ↦ 0))
159		fconstmpt 5721	. . 3 ⊢ ((1...6) × {0}) = (𝑚 ∈ (1...6) ↦ 0)
160		re0g 21824	. . . . 5 ⊢ 0 = (0_g‘ℝ_fld)
161	54, 160	frlm0 21966	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ (1...6) ∈ V) → ((1...6) × {0}) = (0_g‘(ℝ_fld freeLMod (1...6))))
162	97, 19, 161	mp2an 705	. . 3 ⊢ ((1...6) × {0}) = (0_g‘(ℝ_fld freeLMod (1...6)))
163	159, 162	eqtr3i 2787	. 2 ⊢ (𝑚 ∈ (1...6) ↦ 0) = (0_g‘(ℝ_fld freeLMod (1...6)))
164	158, 163	eqtrdi 2813	1 ⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = (0_g‘(ℝ_fld freeLMod (1...6))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 Vcvv 3453 ⦋csb 3850 ∖ cdif 3899 ∅c0 4282 {csn 4587 class class class wbr 5107 ↦ cmpt 5190 × cxp 5657 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 ∘_f cof 7679 tpos ctpos 8226 curry ccur 8266 ↑_m cmap 8829 Fincfn 8955 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 ≤ cle 11269 ℕcn 12258 2c2 12320 3c3 12321 4c4 12322 5c5 12323 6c6 12324 ...cfz 13561 ↑cexp 14125 Basecbs 17303 ·_𝑠 cvsca 17348 0_gc0g 17526 Σ_g cgsu 17527 Ringcrg 20371 CRingccrg 20372 DivRingcdr 20889 Fieldcfield 20890 ℝ_fldcrefld 21816 freeLMod cfrlm 21958 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 ax-addf 11204 ax-mulf 11205

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-rmo 3367 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-tp 4592 df-op 4594 df-uni 4871 df-int 4911 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-se 5613 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-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-cur 8268 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-0g 17528 df-gsum 17529 df-prds 17534 df-pws 17536 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-mhm 18890 df-grp 19059 df-minusg 19060 df-sbg 19061 df-subg 19245 df-cntz 19443 df-cmn 19908 df-abl 19909 df-mgp 20273 df-rng 20287 df-ur 20320 df-ring 20373 df-cring 20374 df-oppr 20477 df-dvdsr 20497 df-unit 20498 df-invr 20528 df-dvr 20541 df-subrng 20707 df-subrg 20731 df-drng 20891 df-field 20892 df-lmod 21045 df-lss 21115 df-sra 21356 df-rgmod 21357 df-cnfld 21585 df-refld 21817 df-dsmm 21944 df-frlm 21959 df-veronese 50784

This theorem is used by: veroquadnolindfd 50800

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