Theorem plyrem 26286

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16484). 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 26178	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 482	. . . . . . . 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 26285	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
6	5	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
7	6	simp1d 1143	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
8	6	simp2d 1144	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9		ax-1ne0 11109	. . . . . . . . . 10 ⊢ 1 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
11	8, 10	eqnetrd 3000	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12		fveq2 6844	. . . . . . . . . 10 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13		dgr0 26241	. . . . . . . . . 10 ⊢ (deg‘0𝑝) = 0
14	12, 13	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
15	14	necon3i 2965	. . . . . . . 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 26284	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
19	3, 7, 16, 18	syl3anc 1374	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20		0lt1 11673	. . . . . . . 8 ⊢ 0 < 1
21	20, 8	breqtrrid 5138	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22		fveq2 6844	. . . . . . . . 9 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
23	22, 13	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
24	23	breq1d 5110	. . . . . . 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		pm2.62 900	. . . . . 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 5126	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29		quotcl2 26283	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
30	3, 7, 16, 29	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31		plymulcl 26199	. . . . . . . . 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 26200	. . . . . . . 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 2841	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36		dgrcl 26211	. . . . . 6 ⊢ (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38		nn0lt10b 12568	. . . . 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 232	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41		0dgrb 26224	. . . 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 232	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
44	43	fveq1d 6846	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
45	17	fveq1i 6845	. . . . . . 7 ⊢ (𝑅‘𝐴) = ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴)
46		plyf 26176	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48	47	ffnd 6673	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49		plyf 26176	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
50	7, 49	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
51	50	ffnd 6673	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52		plyf 26176	. . . . . . . . . . . 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 6673	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55		cnex 11121	. . . . . . . . . . 11 ⊢ ℂ ∈ V
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57		inidm 4181	. . . . . . . . . 10 ⊢ (ℂ ∩ ℂ) = ℂ
58	51, 54, 56, 56, 57	offn 7647	. . . . . . . . 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 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡𝐺 “ {0}) = {𝐴})
61		ssun1 4132	. . . . . . . . . . . . . . 15 ⊢ (◡𝐺 “ {0}) ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))
62	60, 61	eqsstrrdi 3981	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
63		snssg 4742	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
64	63	adantl 481	. . . . . . . . . . . . . 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 26262	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
68	65, 67	eleqtrrd 2840	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}))
69		fniniseg 7016	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0))
72	71	simprd 495	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
73	72	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
74	48, 58, 56, 56, 57, 59, 73	ofval 7645	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
75	74	anabss3 676	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
76	45, 75	eqtrid 2784	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((𝐹‘𝐴) − 0))
77	46	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) ∈ ℂ)
78	77	subid1d 11495	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹‘𝐴) − 0) = (𝐹‘𝐴))
79	76, 78	eqtrd 2772	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = (𝐹‘𝐴))
80		fvex 6857	. . . . . . 7 ⊢ (𝑅‘0) ∈ V
81	80	fvconst2 7162	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
82	81	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
83	44, 79, 82	3eqtr3d 2780	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝑅‘0))
84	83	sneqd 4594	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹‘𝐴)} = {(𝑅‘0)})
85	84	xpeq2d 5664	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {(𝑅‘0)}))
86	43, 85	eqtr4d 2775	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  {csn 4582   class class class wbr 5100   × cxp 5632  ^◡ccnv 5633   “ cima 5637   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∘_f cof 7632  ℂcc 11038  0cc0 11040  1c1 11041   · cmul 11045   < clt 11180   − cmin 11378  ℕ₀cn0 12415  0_𝑝c0p 25643  Polycply 26162  X_pcidp 26163  degcdgr 26165   quot cquot 26271

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-pm 8780  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-sup 9359  df-inf 9360  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-rp 12920  df-fz 13438  df-fzo 13585  df-fl 13726  df-seq 13939  df-exp 13999  df-hash 14268  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-clim 15425  df-rlim 15426  df-sum 15624  df-0p 25644  df-ply 26166  df-idp 26167  df-coe 26168  df-dgr 26169  df-quot 26272

This theorem is referenced by:  facth  26287