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Theorem vieta1 26372

Description: The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree 𝑁 has 𝑁 distinct roots, then the sum over these roots can be calculated as -𝐴(𝑁 − 1) / 𝐴(𝑁). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)

**Hypotheses**
Ref	Expression
vieta1.1	⊢ 𝐴 = (coeff‘𝐹)
vieta1.2	⊢ 𝑁 = (deg‘𝐹)
vieta1.3	⊢ 𝑅 = (^◡𝐹 “ {0})
vieta1.4	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
vieta1.5	⊢ (𝜑 → (♯‘𝑅) = 𝑁)
vieta1.6	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
vieta1	⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))

Distinct variable groups: 𝑥,𝑅 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝑆(𝑥) 𝐹(𝑥) 𝑁(𝑥)

Proof of Theorem vieta1 Dummy variables 𝑓 𝑘 𝑦 𝑧 𝑑 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vieta1.5	. 2 ⊢ (𝜑 → (♯‘𝑅) = 𝑁)
2		fveq2 6920	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
3	2	eqeq2d 2751	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑁 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝐹)))
4		cnveq 5898	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ^◡𝑓 = ^◡𝐹)
5	4	imaeq1d 6088	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (^◡𝑓 “ {0}) = (^◡𝐹 “ {0}))
6		vieta1.3	. . . . . . . . 9 ⊢ 𝑅 = (^◡𝐹 “ {0})
7	5, 6	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (^◡𝑓 “ {0}) = 𝑅)
8	7	fveq2d 6924	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (♯‘(^◡𝑓 “ {0})) = (♯‘𝑅))
9		vieta1.2	. . . . . . . 8 ⊢ 𝑁 = (deg‘𝐹)
10	2, 9	eqtr4di 2798	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑁)
11	8, 10	eqeq12d 2756	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((♯‘(^◡𝑓 “ {0})) = (deg‘𝑓) ↔ (♯‘𝑅) = 𝑁))
12	3, 11	anbi12d 631	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁)))
13	9	biantrur 530	. . . . 5 ⊢ ((♯‘𝑅) = 𝑁 ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁))
14	12, 13	bitr4di 289	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (♯‘𝑅) = 𝑁))
15	7	sumeq1d 15748	. . . . 5 ⊢ (𝑓 = 𝐹 → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = Σ𝑥 ∈ 𝑅 𝑥)
16		fveq2 6920	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
17		vieta1.1	. . . . . . . . 9 ⊢ 𝐴 = (coeff‘𝐹)
18	16, 17	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴)
19	10	oveq1d 7463	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) − 1) = (𝑁 − 1))
20	18, 19	fveq12d 6927	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = (𝐴‘(𝑁 − 1)))
21	18, 10	fveq12d 6927	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘(deg‘𝑓)) = (𝐴‘𝑁))
22	20, 21	oveq12d 7466	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = ((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))
23	22	negeqd 11530	. . . . 5 ⊢ (𝑓 = 𝐹 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))
24	15, 23	eqeq12d 2756	. . . 4 ⊢ (𝑓 = 𝐹 → (Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))))
25	14, 24	imbi12d 344	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((♯‘𝑅) = 𝑁 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))))
26		vieta1.6	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27		eqeq1 2744	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑦 = (deg‘𝑓) ↔ 1 = (deg‘𝑓)))
28	27	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 1 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))))
29	28	imbi1d 341	. . . . . 6 ⊢ (𝑦 = 1 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
30	29	ralbidv 3184	. . . . 5 ⊢ (𝑦 = 1 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
31		eqeq1 2744	. . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (deg‘𝑓) ↔ 𝑑 = (deg‘𝑓)))
32	31	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 𝑑 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))))
33	32	imbi1d 341	. . . . . 6 ⊢ (𝑦 = 𝑑 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
34	33	ralbidv 3184	. . . . 5 ⊢ (𝑦 = 𝑑 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
35		eqeq1 2744	. . . . . . . 8 ⊢ (𝑦 = (𝑑 + 1) → (𝑦 = (deg‘𝑓) ↔ (𝑑 + 1) = (deg‘𝑓)))
36	35	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = (𝑑 + 1) → ((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))))
37	36	imbi1d 341	. . . . . 6 ⊢ (𝑦 = (𝑑 + 1) → (((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
38	37	ralbidv 3184	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
39		eqeq1 2744	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑦 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝑓)))
40	39	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))))
41	40	imbi1d 341	. . . . . 6 ⊢ (𝑦 = 𝑁 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
42	41	ralbidv 3184	. . . . 5 ⊢ (𝑦 = 𝑁 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
43		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
44	43	coef3 26291	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (Poly‘ℂ) → (coeff‘𝑓):ℕ₀⟶ℂ)
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (coeff‘𝑓):ℕ₀⟶ℂ)
46		0nn0 12568	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
47		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑓):ℕ₀⟶ℂ ∧ 0 ∈ ℕ₀) → ((coeff‘𝑓)‘0) ∈ ℂ)
48	45, 46, 47	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) ∈ ℂ)
49		1nn0 12569	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
50		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑓):ℕ₀⟶ℂ ∧ 1 ∈ ℕ₀) → ((coeff‘𝑓)‘1) ∈ ℂ)
51	45, 49, 50	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ∈ ℂ)
52		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 = (deg‘𝑓))
53	52	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) = ((coeff‘𝑓)‘(deg‘𝑓)))
54		ax-1ne0 11253	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 0
55	54	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 ≠ 0)
56	52, 55	eqnetrrd 3015	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (deg‘𝑓) ≠ 0)
57		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 0_𝑝 → (deg‘𝑓) = (deg‘0_𝑝))
58		dgr0 26322	. . . . . . . . . . . . . . . . 17 ⊢ (deg‘0_𝑝) = 0
59	57, 58	eqtrdi 2796	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 0_𝑝 → (deg‘𝑓) = 0)
60	59	necon3i 2979	. . . . . . . . . . . . . . 15 ⊢ ((deg‘𝑓) ≠ 0 → 𝑓 ≠ 0_𝑝)
61	56, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 ≠ 0_𝑝)
62		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (deg‘𝑓) = (deg‘𝑓)
63	62, 43	dgreq0 26325	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (Poly‘ℂ) → (𝑓 = 0_𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
64	63	necon3bid 2991	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℂ) → (𝑓 ≠ 0_𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
65	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 ≠ 0_𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
66	61, 65	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
67	53, 66	eqnetrd 3014	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ≠ 0)
68	48, 51, 67	divcld 12070	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
69	68	negcld 11634	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
70		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) → 𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
71	70	sumsn 15794	. . . . . . . . . 10 ⊢ ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
72	69, 69, 71	syl2anc 583	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
73	72	adantrr 716	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
74		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (^◡𝑓 “ {0}) = (^◡𝑓 “ {0})
75	74	fta1 26368	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 𝑓 ≠ 0_𝑝) → ((^◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑓 “ {0})) ≤ (deg‘𝑓)))
76	61, 75	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((^◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑓 “ {0})) ≤ (deg‘𝑓)))
77	76	simpld 494	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (^◡𝑓 “ {0}) ∈ Fin)
78	77	adantrr 716	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → (^◡𝑓 “ {0}) ∈ Fin)
79	43, 62	coeid2 26298	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
80	69, 79	syldan 590	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
81	52	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0...1) = (0...(deg‘𝑓)))
82	81	sumeq1d 15748	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
83		nn0uz 12945	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ = (ℤ_≥‘0)
84		1e0p1 12800	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
85		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘1))
86		oveq2 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))
87	85, 86	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)))
88	45	ffvelcdmda 7118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝑓)‘𝑘) ∈ ℂ)
89		expcl 14130	. . . . . . . . . . . . . . . . . 18 ⊢ ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
90	69, 89	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ₀) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
91	88, 90	mulcld 11310	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ₀) → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) ∈ ℂ)
92		0z 12650	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
93	69	exp0d 14190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0) = 1)
94	93	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = (((coeff‘𝑓)‘0) · 1))
95	48	mulridd 11307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · 1) = ((coeff‘𝑓)‘0))
96	94, 95	eqtrd 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = ((coeff‘𝑓)‘0))
97	96, 48	eqeltrd 2844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ)
98		fveq2 6920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘0))
99		oveq2 7456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))
100	98, 99	oveq12d 7466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
101	100	fsum1 15795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
102	92, 97, 101	sylancr 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
103	102, 96	eqtrd 2780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0))
104	103, 46	jctil 519	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0)))
105	69	exp1d 14191	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1) = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
106	105	oveq2d 7464	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))))
107	51, 68	mulneg2d 11744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))))
108	48, 51, 67	divcan2d 12072	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = ((coeff‘𝑓)‘0))
109	108	negeqd 11530	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -((coeff‘𝑓)‘0))
110	106, 107, 109	3eqtrd 2784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = -((coeff‘𝑓)‘0))
111	110	oveq2d 7464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)))
112	48	negidd 11637	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)) = 0)
113	111, 112	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = 0)
114	83, 84, 87, 91, 104, 113	fsump1i 15817	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0))
115	114	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0)
116	80, 82, 115	3eqtr2d 2786	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)
117		plyf 26257	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
118	117	ffnd 6748	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓 Fn ℂ)
119	118	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 Fn ℂ)
120		fniniseg 7093	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn ℂ → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (^◡𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)))
121	119, 120	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (^◡𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)))
122	69, 116, 121	mpbir2and 712	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (^◡𝑓 “ {0}))
123	122	snssd 4834	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (^◡𝑓 “ {0}))
124	123	adantrr 716	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (^◡𝑓 “ {0}))
125		hashsng 14418	. . . . . . . . . . . . . . 15 ⊢ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
126	69, 125	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
127	126, 52	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
128	127	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
129		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))
130	128, 129	eqtr4d 2783	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(^◡𝑓 “ {0})))
131		snfi 9109	. . . . . . . . . . . . 13 ⊢ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin
132		hashen 14396	. . . . . . . . . . . . 13 ⊢ (({-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin ∧ (^◡𝑓 “ {0}) ∈ Fin) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(^◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (^◡𝑓 “ {0})))
133	131, 77, 132	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(^◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (^◡𝑓 “ {0})))
134	133	adantrr 716	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(^◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (^◡𝑓 “ {0})))
135	130, 134	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (^◡𝑓 “ {0}))
136		fisseneq 9320	. . . . . . . . . 10 ⊢ (((^◡𝑓 “ {0}) ∈ Fin ∧ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (^◡𝑓 “ {0}) ∧ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (^◡𝑓 “ {0})) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (^◡𝑓 “ {0}))
137	78, 124, 135, 136	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (^◡𝑓 “ {0}))
138	137	sumeq1d 15748	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥)
139		1m1e0 12365	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
140	52	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 − 1) = ((deg‘𝑓) − 1))
141	139, 140	eqtr3id 2794	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 0 = ((deg‘𝑓) − 1))
142	141	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
143	142, 53	oveq12d 7466	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
144	143	negeqd 11530	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
145	144	adantrr 716	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
146	73, 138, 145	3eqtr3d 2788	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
147	146	ex 412	. . . . . 6 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
148	147	rgen 3069	. . . . 5 ⊢ ∀𝑓 ∈ (Poly‘ℂ)((1 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
149		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
150	149	cbvsumv 15744	. . . . . . . . . . 11 ⊢ Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥
151	150	eqeq1i 2745	. . . . . . . . . 10 ⊢ (Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
152	151	imbi2i 336	. . . . . . . . 9 ⊢ (((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
153	152	ralbii 3099	. . . . . . . 8 ⊢ (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
154		eqid 2740	. . . . . . . . . 10 ⊢ (coeff‘𝑔) = (coeff‘𝑔)
155		eqid 2740	. . . . . . . . . 10 ⊢ (deg‘𝑔) = (deg‘𝑔)
156		eqid 2740	. . . . . . . . . 10 ⊢ (^◡𝑔 “ {0}) = (^◡𝑔 “ {0})
157		simp1r 1198	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → 𝑔 ∈ (Poly‘ℂ))
158		simp3r 1202	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))
159		simp1l 1197	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → 𝑑 ∈ ℕ)
160		simp3l 1201	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → (𝑑 + 1) = (deg‘𝑔))
161		simp2 1137	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
162	161, 153	sylib 218	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
163		eqid 2740	. . . . . . . . . 10 ⊢ (𝑔 quot (X_p ∘_f − (ℂ × {𝑧}))) = (𝑔 quot (X_p ∘_f − (ℂ × {𝑧})))
164	154, 155, 156, 157, 158, 159, 160, 162, 163	vieta1lem2 26371	. . . . . . . . 9 ⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔))) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))
165	164	3exp 1119	. . . . . . . 8 ⊢ ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (^◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
166	153, 165	biimtrrid 243	. . . . . . 7 ⊢ ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
167	166	ralrimdva 3160	. . . . . 6 ⊢ (𝑑 ∈ ℕ → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑔 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
168		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
169	168	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → ((𝑑 + 1) = (deg‘𝑔) ↔ (𝑑 + 1) = (deg‘𝑓)))
170		cnveq 5898	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ^◡𝑔 = ^◡𝑓)
171	170	imaeq1d 6088	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (^◡𝑔 “ {0}) = (^◡𝑓 “ {0}))
172	171	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (♯‘(^◡𝑔 “ {0})) = (♯‘(^◡𝑓 “ {0})))
173	172, 168	eqeq12d 2756	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → ((♯‘(^◡𝑔 “ {0})) = (deg‘𝑔) ↔ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)))
174	169, 173	anbi12d 631	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓))))
175	171	sumeq1d 15748	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥)
176		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (coeff‘𝑔) = (coeff‘𝑓))
177	168	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) − 1) = ((deg‘𝑓) − 1))
178	176, 177	fveq12d 6927	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((coeff‘𝑔)‘((deg‘𝑔) − 1)) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
179	176, 168	fveq12d 6927	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((coeff‘𝑔)‘(deg‘𝑔)) = ((coeff‘𝑓)‘(deg‘𝑓)))
180	178, 179	oveq12d 7466	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
181	180	negeqd 11530	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
182	175, 181	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → (Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) ↔ Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
183	174, 182	imbi12d 344	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ((((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
184	183	cbvralvw 3243	. . . . . 6 ⊢ (∀𝑔 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(^◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (^◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
185	167, 184	imbitrdi 251	. . . . 5 ⊢ (𝑑 ∈ ℕ → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
186	30, 34, 38, 42, 148, 185	nnind 12311	. . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
187	26, 186	syl 17	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(^◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (^◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
188		plyssc 26259	. . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
189		vieta1.4	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
190	188, 189	sselid 4006	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
191	25, 187, 190	rspcdva 3636	. 2 ⊢ (𝜑 → ((♯‘𝑅) = 𝑁 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))))
192	1, 191	mpd 15	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ⊆ wss 3976 {csn 4648 class class class wbr 5166 × cxp 5698 ^◡ccnv 5699 “ cima 5703 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∘_f cof 7712 ≈ cen 9000 Fincfn 9003 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ...cfz 13567 ↑cexp 14112 ♯chash 14379 Σcsu 15734 0_𝑝c0p 25723 Polycply 26243 X_pcidp 26244 coeffccoe 26245 degcdgr 26246 quot cquot 26350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-0p 25724 df-ply 26247 df-idp 26248 df-coe 26249 df-dgr 26250 df-quot 26351

This theorem is referenced by: basellem5 27146

W3C validator