veroquadmodzerod - Mathbox for Jiamin Zhao

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Theorem veroquadmodzerod 50984

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 6710	. . . . . 6 ⊢ (𝜑 → 𝐾 Fn (1...6))
3		veroquad.a	. . . . . . . . . . . 12 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
4		2fveq3 6890	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
5	4	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
6		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
7	5, 6	cbvmpov 7515	. . . . . . . . . . . 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 2783	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
9	8	tposmpo 8280	. . . . . . . . . 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 7085	. . . . . . . . . 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 50972	. . . . . . . . . 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 8084	. . . . . . . 8 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
19		ovex 7453	. . . . . . . . . 10 ⊢ (1...6) ∈ V
20		1nn 12346	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
21		6nn 12432	. . . . . . . . . . . 12 ⊢ 6 ∈ ℕ
22		1re 11308	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
23		6re 12433	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℝ
24		1lt6 12530	. . . . . . . . . . . . 13 ⊢ 1 < 6
25	22, 23, 24	ltleii 11433	. . . . . . . . . . . 12 ⊢ 1 ≤ 6
26		elfz1b 13727	. . . . . . . . . . . 12 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
27	20, 21, 25, 26	mpbir3an 1360	. . . . . . . . . . 11 ⊢ 1 ∈ (1...6)
28	27	ne0ii 4289	. . . . . . . . . 10 ⊢ (1...6) ≠ ∅
29		eldifsn 4747	. . . . . . . . . 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 11291	. . . . . . . . 9 ⊢ ℝ ∈ V
33	32	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
34		curf 8890	. . . . . . . 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 6710	. . . . . 6 ⊢ (𝜑 → curry tpos 𝑉 Fn (1...6))
37	19	a1i 11	. . . . . 6 ⊢ (𝜑 → (1...6) ∈ V)
38		inidm 4123	. . . . . 6 ⊢ ((1...6) ∩ (1...6)) = (1...6)
39		eqidd 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) = (𝐾‘𝑛))
40	17	ralrimivva 3205	. . . . . . . . . 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 8287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)))
46		csbfv 6932	. . . . . . . . 9 ⊢ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣) = ((veronese‘(𝐴‘𝑢))‘𝑛)
47	46	mpteq2i 5200	. . . . . . . 8 ⊢ (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛))
48	45, 47	eqtrdi 2811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)))
49		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
50	49	fveq1d 6887	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → ((veronese‘(𝐴‘𝑢))‘𝑛) = ((veronese‘(𝐴‘𝑖))‘𝑛))
51	50	cbvmptv 5208	. . . . . . 7 ⊢ (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
52	48, 51	eqtrdi 2811	. . . . . 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 7702	. . . . 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 2760	. . . . . . 7 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
55		eqid 2760	. . . . . . 7 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
56		rebase 21912	. . . . . . 7 ⊢ ℝ = (Base‘ℝ_fld)
57	1	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ ℝ)
58		refld 21925	. . . . . . . . . . . . 13 ⊢ ℝ_fld ∈ Field
59	58	elexi 3472	. . . . . . . . . . . 12 ⊢ ℝ_fld ∈ V
60		fzfi 14115	. . . . . . . . . . . 12 ⊢ (1...6) ∈ Fin
61	54, 56	frlmfibas 22068	. . . . . . . . . . . 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 6696	. . . . . . . . 9 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(Base‘(ℝ_fld freeLMod (1...6))))
65	64	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
66	52, 65	eqeltrrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
67		eqid 2760	. . . . . . 7 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
68		remulr 21917	. . . . . . 7 ⊢ · = (._r‘ℝ_fld)
69	54, 55, 56, 43, 57, 66, 67, 68	frlmvscafval 22072	. . . . . 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 5197	. . . . 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 6900	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ V)
72		fnconstg 6770	. . . . . . . 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 6898	. . . . . . . . 9 ⊢ ((veronese‘(𝐴‘𝑖))‘𝑛) ∈ V
75		eqid 2760	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
76	74, 75	fnmpti 6682	. . . . . . . 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 6898	. . . . . . . . 9 ⊢ (𝐾‘𝑛) ∈ V
80	79	fvconst2 7210	. . . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑚)))
83	82	fveq1d 6887	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
84		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → 𝑚 ∈ (1...6))
85		fvexd 6900	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) ∈ V)
86	75, 83, 84, 85	fvmptd3 7017	. . . . . . . 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 7702	. . . . . 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 5197	. . . . 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 2799	. . . 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 7436	. . 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 2760	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
93		isfld 20993	. . . . . . . 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 20987	. . . . . 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 7085	. . . . . . . 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 50972	. . . . . . . 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 11339	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)) ∈ ℝ)
107	106	fmpttd 7115	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))):(1...6)⟶ℝ)
108	32, 19	elmap 8899	. . . . 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 2760	. . . . 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 6900	. . . . 5 ⊢ (𝜑 → (0_g‘(ℝ_fld freeLMod (1...6))) ∈ V)
113	110, 111, 109, 112	fsuppmptdm 9368	. . . 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 22078	. . 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 50981	. . . . . . . . . . . . 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 6900	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (veronese‘(𝐴‘𝑚)) ∈ V)
118	82, 116, 78, 117	fvmptd4 7018	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (curry 𝑉‘𝑚) = (veronese‘(𝐴‘𝑚)))
119	118	fveq1d 6887	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((curry 𝑉‘𝑚)‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
120	119	eqcomd 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) = ((curry 𝑉‘𝑚)‘𝑛))
121	120	oveq2d 7436	. . . . . . . 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 5197	. . . . . 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 7436	. . . . 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 6887	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑗))
126		fveq2 6885	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → ((veronese‘(𝐴‘𝑚))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑛))
127	125, 126	cbvmpov 7515	. . . . . . 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 2783	. . . . . 6 ⊢ 𝑉 = (𝑚 ∈ (1...6), 𝑛 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑚))‘𝑛))
129		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (𝐴‘𝑖) = (𝐴‘𝑚))
130	129	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘1) = ((𝐴‘𝑚)‘1))
131	130	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1)↑2) = (((𝐴‘𝑚)‘1)↑2))
132	131	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) = ((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)))
133	129	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘2) = ((𝐴‘𝑚)‘2))
134	133	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2)↑2) = (((𝐴‘𝑚)‘2)↑2))
135	134	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) = ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2)))
136	132, 135	oveq12d 7438	. . . . . . . . . 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 6887	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘3) = ((𝐴‘𝑚)‘3))
138	137	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3)↑2) = (((𝐴‘𝑚)‘3)↑2))
139	138	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) = ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2)))
140	136, 139	oveq12d 7438	. . . . . . . . 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 7438	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)) = (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2)))
142	141	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) = ((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))))
143	133, 137	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)) = (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))
144	143	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) = ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3))))
145	142, 144	oveq12d 7438	. . . . . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)) = (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1)))
147	146	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) = ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))
148	145, 147	oveq12d 7438	. . . . . . . . 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 7438	. . . . . . . 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 2762	. . . . . . 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 3154	. . . . . . . 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 3577	. . . . . 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 50983	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))) = 0)
156	124, 155	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) = 0)
157	156	mpteq2dva 5197	. . 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 2799	. 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 5712	. . 3 ⊢ ((1...6) × {0}) = (𝑚 ∈ (1...6) ↦ 0)
160		re0g 21918	. . . . 5 ⊢ 0 = (0_g‘ℝ_fld)
161	54, 160	frlm0 22060	. . . 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 2785	. 2 ⊢ (𝑚 ∈ (1...6) ↦ 0) = (0_g‘(ℝ_fld freeLMod (1...6)))
164	158, 163	eqtrdi 2811	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 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 Vcvv 3450 ⦋csb 3846 ∖ cdif 4045 ∅c0 4278 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ∘_f cof 7691 tpos ctpos 8242 curry ccur 8282 ↑_m cmap 8847 Fincfn 8973 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ≤ cle 11344 ℕcn 12335 2c2 12397 3c3 12398 4c4 12399 5c5 12400 6c6 12401 ...cfz 13639 ↑cexp 14204 Basecbs 17387 ·_𝑠 cvsca 17432 0_gc0g 17610 Σ_g cgsu 17611 Ringcrg 20459 CRingccrg 20460 DivRingcdr 20980 Fieldcfield 20981 ℝ_fldcrefld 21910 freeLMod cfrlm 22052 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 ax-mulf 11280

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-int 4907 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-se 5604 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-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 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-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 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-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 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-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-grp 19147 df-minusg 19148 df-sbg 19149 df-subg 19333 df-cntz 19531 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-lmod 21137 df-lss 21207 df-sra 21448 df-rgmod 21449 df-cnfld 21679 df-refld 21911 df-dsmm 22038 df-frlm 22053 df-veronese 50969

This theorem is used by: veroquadnolindfd 50985

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