veroquadmodzerod - Mathbox for Jiamin Zhao

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

Theorem veroquadmodzerod 51262

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 6729	. . . . . 6 ⊢ (𝜑 → 𝐾 Fn (1...6))
3		veroquad.a	. . . . . . . . . . . 12 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
4		2fveq3 6910	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
5	4	fveq1d 6907	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
6		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
7	5, 6	cbvmpov 7535	. . . . . . . . . . . 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 2786	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
9	8	tposmpo 8305	. . . . . . . . . 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 7106	. . . . . . . . . 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 51250	. . . . . . . . . 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 8106	. . . . . . . 8 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
19		ovex 7473	. . . . . . . . . 10 ⊢ (1...6) ∈ V
20		1nn 12370	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
21		6nn 12456	. . . . . . . . . . . 12 ⊢ 6 ∈ ℕ
22		1re 11332	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
23		6re 12457	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℝ
24		1lt6 12554	. . . . . . . . . . . . 13 ⊢ 1 < 6
25	22, 23, 24	ltleii 11457	. . . . . . . . . . . 12 ⊢ 1 ≤ 6
26		elfz1b 13751	. . . . . . . . . . . 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 4748	. . . . . . . . . 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 11315	. . . . . . . . 9 ⊢ ℝ ∈ V
33	32	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
34		curf 8915	. . . . . . . 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 1399	. . . . . . 7 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
36	35	ffnd 6729	. . . . . 6 ⊢ (𝜑 → curry tpos 𝑉 Fn (1...6))
37	19	a1i 11	. . . . . 6 ⊢ (𝜑 → (1...6) ∈ V)
38		inidm 4126	. . . . . 6 ⊢ ((1...6) ∩ (1...6)) = (1...6)
39		eqidd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) = (𝐾‘𝑛))
40	17	ralrimivva 3209	. . . . . . . . . 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 8312	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)))
46		csbfv 6952	. . . . . . . . 9 ⊢ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣) = ((veronese‘(𝐴‘𝑢))‘𝑛)
47	46	mpteq2i 5204	. . . . . . . 8 ⊢ (𝑢 ∈ (1...6) ↦ ⦋𝑛 / 𝑣⦌((veronese‘(𝐴‘𝑢))‘𝑣)) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛))
48	45, 47	eqtrdi 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) = (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)))
49		2fveq3 6910	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
50	49	fveq1d 6907	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → ((veronese‘(𝐴‘𝑢))‘𝑛) = ((veronese‘(𝐴‘𝑖))‘𝑛))
51	50	cbvmptv 5212	. . . . . . 7 ⊢ (𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
52	48, 51	eqtrdi 2814	. . . . . 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 7723	. . . . 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 2763	. . . . . . 7 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
55		eqid 2763	. . . . . . 7 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
56		rebase 21939	. . . . . . 7 ⊢ ℝ = (Base‘ℝ_fld)
57	1	ffvelcdmda 7105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ ℝ)
58		refld 21952	. . . . . . . . . . . . 13 ⊢ ℝ_fld ∈ Field
59	58	elexi 3476	. . . . . . . . . . . 12 ⊢ ℝ_fld ∈ V
60		fzfi 14139	. . . . . . . . . . . 12 ⊢ (1...6) ∈ Fin
61	54, 56	frlmfibas 22095	. . . . . . . . . . . 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 6715	. . . . . . . . 9 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(Base‘(ℝ_fld freeLMod (1...6))))
65	64	ffvelcdmda 7105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (curry tpos 𝑉‘𝑛) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
66	52, 65	eqeltrrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) ∈ (Base‘(ℝ_fld freeLMod (1...6))))
67		eqid 2763	. . . . . . 7 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
68		remulr 21944	. . . . . . 7 ⊢ · = (._r‘ℝ_fld)
69	54, 55, 56, 43, 57, 66, 67, 68	frlmvscafval 22099	. . . . . 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 5201	. . . . 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 6920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝐾‘𝑛) ∈ V)
72		fnconstg 6789	. . . . . . . 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 6918	. . . . . . . . 9 ⊢ ((veronese‘(𝐴‘𝑖))‘𝑛) ∈ V
75		eqid 2763	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛)) = (𝑖 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑛))
76	74, 75	fnmpti 6701	. . . . . . . 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 6918	. . . . . . . . 9 ⊢ (𝐾‘𝑛) ∈ V
80	79	fvconst2 7230	. . . . . . . 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 6910	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑚)))
83	82	fveq1d 6907	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
84		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → 𝑚 ∈ (1...6))
85		fvexd 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) ∈ V)
86	75, 83, 84, 85	fvmptd3 7037	. . . . . . . 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 7723	. . . . . 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 5201	. . . . 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 2802	. . . 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 7456	. . 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 2763	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
93		isfld 21020	. . . . . . . 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 21014	. . . . . 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 7106	. . . . . . . 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 51250	. . . . . . . 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 11363	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)) ∈ ℝ)
107	106	fmpttd 7136	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...6)) → (𝑚 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛))):(1...6)⟶ℝ)
108	32, 19	elmap 8924	. . . . 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 2763	. . . . 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 6920	. . . . 5 ⊢ (𝜑 → (0_g‘(ℝ_fld freeLMod (1...6))) ∈ V)
113	110, 111, 109, 112	fsuppmptdm 9392	. . . 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 22105	. . 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 51259	. . . . . . . . . . . . 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 6920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (veronese‘(𝐴‘𝑚)) ∈ V)
118	82, 116, 78, 117	fvmptd4 7038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → (curry 𝑉‘𝑚) = (veronese‘(𝐴‘𝑚)))
119	118	fveq1d 6907	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((curry 𝑉‘𝑚)‘𝑛) = ((veronese‘(𝐴‘𝑚))‘𝑛))
120	119	eqcomd 2769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...6)) ∧ 𝑚 ∈ (1...6)) → ((veronese‘(𝐴‘𝑚))‘𝑛) = ((curry 𝑉‘𝑚)‘𝑛))
121	120	oveq2d 7456	. . . . . . . 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 5201	. . . . . 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 7456	. . . . 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 6907	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑗))
126		fveq2 6905	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → ((veronese‘(𝐴‘𝑚))‘𝑗) = ((veronese‘(𝐴‘𝑚))‘𝑛))
127	125, 126	cbvmpov 7535	. . . . . . 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 2786	. . . . . 6 ⊢ 𝑉 = (𝑚 ∈ (1...6), 𝑛 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑚))‘𝑛))
129		fveq2 6905	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (𝐴‘𝑖) = (𝐴‘𝑚))
130	129	fveq1d 6907	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘1) = ((𝐴‘𝑚)‘1))
131	130	oveq1d 7455	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1)↑2) = (((𝐴‘𝑚)‘1)↑2))
132	131	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) = ((𝐾‘1) · (((𝐴‘𝑚)‘1)↑2)))
133	129	fveq1d 6907	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘2) = ((𝐴‘𝑚)‘2))
134	133	oveq1d 7455	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2)↑2) = (((𝐴‘𝑚)‘2)↑2))
135	134	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2)) = ((𝐾‘2) · (((𝐴‘𝑚)‘2)↑2)))
136	132, 135	oveq12d 7458	. . . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → ((𝐴‘𝑖)‘3) = ((𝐴‘𝑚)‘3))
138	137	oveq1d 7455	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3)↑2) = (((𝐴‘𝑚)‘3)↑2))
139	138	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2)) = ((𝐾‘3) · (((𝐴‘𝑚)‘3)↑2)))
140	136, 139	oveq12d 7458	. . . . . . . . 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 7458	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)) = (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2)))
142	141	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) = ((𝐾‘4) · (((𝐴‘𝑚)‘1) · ((𝐴‘𝑚)‘2))))
143	133, 137	oveq12d 7458	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)) = (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3)))
144	143	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3))) = ((𝐾‘5) · (((𝐴‘𝑚)‘2) · ((𝐴‘𝑚)‘3))))
145	142, 144	oveq12d 7458	. . . . . . . . . 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 7458	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1)) = (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1)))
147	146	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))) = ((𝐾‘6) · (((𝐴‘𝑚)‘3) · ((𝐴‘𝑚)‘1))))
148	145, 147	oveq12d 7458	. . . . . . . . 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 7458	. . . . . . . 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 2765	. . . . . . 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 3158	. . . . . . . 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 3581	. . . . . 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 51261	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((curry 𝑉‘𝑚)‘𝑛)))) = 0)
156	124, 155	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...6)) → (ℝ_fld Σ_g (𝑛 ∈ (1...6) ↦ ((𝐾‘𝑛) · ((veronese‘(𝐴‘𝑚))‘𝑛)))) = 0)
157	156	mpteq2dva 5201	. . 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 2802	. 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 5717	. . 3 ⊢ ((1...6) × {0}) = (𝑚 ∈ (1...6) ↦ 0)
160		re0g 21945	. . . . 5 ⊢ 0 = (0_g‘ℝ_fld)
161	54, 160	frlm0 22087	. . . 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 2788	. 2 ⊢ (𝑚 ∈ (1...6) ↦ 0) = (0_g‘(ℝ_fld freeLMod (1...6)))
164	158, 163	eqtrdi 2814	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 2147 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⦋csb 3851 ∖ cdif 4048 ∅c0 4281 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 × cxp 5653 Fn wfn 6550 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 ∘_f cof 7712 tpos ctpos 8267 curry ccur 8307 ↑_m cmap 8872 Fincfn 8998 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 ≤ cle 11368 ℕcn 12359 2c2 12421 3c3 12422 4c4 12423 5c5 12424 6c6 12425 ...cfz 13663 ↑cexp 14228 Basecbs 17413 ·_𝑠 cvsca 17458 0_gc0g 17636 Σ_g cgsu 17637 Ringcrg 20486 CRingccrg 20487 DivRingcdr 21007 Fieldcfield 21008 ℝ_fldcrefld 21937 freeLMod cfrlm 22079 veronesecveronese 51246

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-addf 11303 ax-mulf 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 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 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-cur 8309 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-grp 19173 df-minusg 19174 df-sbg 19175 df-subg 19359 df-cntz 19558 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-subrng 20825 df-subrg 20849 df-drng 21009 df-field 21010 df-lmod 21164 df-lss 21234 df-sra 21475 df-rgmod 21476 df-cnfld 21706 df-refld 21938 df-dsmm 22065 df-frlm 22080 df-veronese 51247

This theorem is used by: veroquadnolindfd 51263

W3C validator