Theorem plyrem 25570

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16350). If a polynomial 𝐹 is divided by the linear factor 𝑥 − 𝐴, the remainder is equal to 𝐹(𝐴), the evaluation of the polynomial at 𝐴 (interpreted as a constant polynomial). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
plyrem.1	⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))
plyrem.2	⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))

Assertion

Ref	Expression
plyrem	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

Proof of Theorem plyrem

Step	Hyp	Ref	Expression
1		plyssc 25466	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 484	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3933	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4		plyrem.1	. . . . . . . . . 10 ⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))
5	4	plyremlem 25569	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
6	5	adantl 483	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
7	6	simp1d 1142	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
8	6	simp2d 1143	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9		ax-1ne0 11045	. . . . . . . . . 10 ⊢ 1 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
11	8, 10	eqnetrd 3009	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12		fveq2 6829	. . . . . . . . . 10 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13		dgr0 25528	. . . . . . . . . 10 ⊢ (deg‘0𝑝) = 0
14	12, 13	eqtrdi 2793	. . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
15	14	necon3i 2974	. . . . . . . 8 ⊢ ((deg‘𝐺) ≠ 0 → 𝐺 ≠ 0𝑝)
16	11, 15	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ≠ 0𝑝)
17		plyrem.2	. . . . . . . 8 ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))
18	17	quotdgr 25568	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
19	3, 7, 16, 18	syl3anc 1371	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20		0lt1 11602	. . . . . . . 8 ⊢ 0 < 1
21	20, 8	breqtrrid 5134	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22		fveq2 6829	. . . . . . . . 9 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
23	22, 13	eqtrdi 2793	. . . . . . . 8 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
24	23	breq1d 5106	. . . . . . 7 ⊢ (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
25	21, 24	syl5ibrcom 247	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26		pm2.62 898	. . . . . 6 ⊢ ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) → ((𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)) → (deg‘𝑅) < (deg‘𝐺)))
27	19, 25, 26	sylc 65	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < (deg‘𝐺))
28	27, 8	breqtrd 5122	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29		quotcl2 25567	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
30	3, 7, 16, 29	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31		plymulcl 25487	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
32	7, 30, 31	syl2anc 585	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33		plysubcl 25488	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
34	3, 32, 33	syl2anc 585	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
35	17, 34	eqeltrid 2842	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36		dgrcl 25499	. . . . . 6 ⊢ (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38		nn0lt10b 12487	. . . . 5 ⊢ ((deg‘𝑅) ∈ ℕ0 → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
39	37, 38	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
40	28, 39	mpbid 231	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41		0dgrb 25512	. . . 4 ⊢ (𝑅 ∈ (Poly‘ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
42	35, 41	syl 17	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
43	40, 42	mpbid 231	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
44	43	fveq1d 6831	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
45	17	fveq1i 6830	. . . . . . 7 ⊢ (𝑅‘𝐴) = ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴)
46		plyf 25464	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47	46	adantr 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48	47	ffnd 6656	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49		plyf 25464	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
50	7, 49	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
51	50	ffnd 6656	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52		plyf 25464	. . . . . . . . . . . 12 ⊢ ((𝐹 quot 𝐺) ∈ (Poly‘ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
53	30, 52	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
54	53	ffnd 6656	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55		cnex 11057	. . . . . . . . . . 11 ⊢ ℂ ∈ V
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57		inidm 4169	. . . . . . . . . 10 ⊢ (ℂ ∩ ℂ) = ℂ
58	51, 54, 56, 56, 57	offn 7612	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) Fn ℂ)
59		eqidd 2738	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝐹‘𝐴))
60	6	simp3d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡𝐺 “ {0}) = {𝐴})
61		ssun1 4123	. . . . . . . . . . . . . . 15 ⊢ (◡𝐺 “ {0}) ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))
62	60, 61	eqsstrrdi 3990	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
63		snssg 4735	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
64	63	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
65	62, 64	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
66		ofmulrt 25547	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∈ V ∧ 𝐺:ℂ⟶ℂ ∧ (𝐹 quot 𝐺):ℂ⟶ℂ) → (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
67	56, 50, 53, 66	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
68	65, 67	eleqtrrd 2841	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}))
69		fniniseg 6997	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∘f · (𝐹 quot 𝐺)) Fn ℂ → (𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)))
70	58, 69	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)))
71	68, 70	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0))
72	71	simprd 497	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
73	72	adantr 482	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
74	48, 58, 56, 56, 57, 59, 73	ofval 7610	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
75	74	anabss3 673	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
76	45, 75	eqtrid 2789	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((𝐹‘𝐴) − 0))
77	46	ffvelcdmda 7021	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) ∈ ℂ)
78	77	subid1d 11426	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹‘𝐴) − 0) = (𝐹‘𝐴))
79	76, 78	eqtrd 2777	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = (𝐹‘𝐴))
80		fvex 6842	. . . . . . 7 ⊢ (𝑅‘0) ∈ V
81	80	fvconst2 7139	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
82	81	adantl 483	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
83	44, 79, 82	3eqtr3d 2785	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝑅‘0))
84	83	sneqd 4589	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹‘𝐴)} = {(𝑅‘0)})
85	84	xpeq2d 5654	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {(𝑅‘0)}))
86	43, 85	eqtr4d 2780	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2941  Vcvv 3442   ∪ cun 3899   ⊆ wss 3901  {csn 4577   class class class wbr 5096   × cxp 5622  ^◡ccnv 5623   “ cima 5627   Fn wfn 6478  ⟶wf 6479  ‘cfv 6483  (class class class)co 7341   ∘_f cof 7597  ℂcc 10974  0cc0 10976  1c1 10977   · cmul 10981   < clt 11114   − cmin 11310  ℕ₀cn0 12338  0_𝑝c0p 24938  Polycply 25450  X_pcidp 25451  degcdgr 25453   quot cquot 25555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-inf2 9502  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053  ax-pre-sup 11054

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-se 5580  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-isom 6492  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7599  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-er 8573  df-map 8692  df-pm 8693  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-sup 9303  df-inf 9304  df-oi 9371  df-card 9800  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-div 11738  df-nn 12079  df-2 12141  df-3 12142  df-n0 12339  df-z 12425  df-uz 12688  df-rp 12836  df-fz 13345  df-fzo 13488  df-fl 13617  df-seq 13827  df-exp 13888  df-hash 14150  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-rlim 15297  df-sum 15497  df-0p 24939  df-ply 25454  df-idp 25455  df-coe 25456  df-dgr 25457  df-quot 25556

This theorem is referenced by:  facth  25571