Theorem plyrem 26365

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16590). 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 26259	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 4006	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4		plyrem.1	. . . . . . . . . 10 ⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))
5	4	plyremlem 26364	. . . . . . . . 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 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 11253	. . . . . . . . . 10 ⊢ 1 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
11	8, 10	eqnetrd 3014	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12		fveq2 6920	. . . . . . . . . 10 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13		dgr0 26322	. . . . . . . . . 10 ⊢ (deg‘0𝑝) = 0
14	12, 13	eqtrdi 2796	. . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
15	14	necon3i 2979	. . . . . . . 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 26363	. . . . . . 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 11812	. . . . . . . 8 ⊢ 0 < 1
21	20, 8	breqtrrid 5204	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22		fveq2 6920	. . . . . . . . 9 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
23	22, 13	eqtrdi 2796	. . . . . . . 8 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
24	23	breq1d 5176	. . . . . . 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 5192	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29		quotcl2 26362	. . . . . . . . . 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 26280	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
32	7, 30, 31	syl2anc 583	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33		plysubcl 26281	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
35	17, 34	eqeltrid 2848	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36		dgrcl 26292	. . . . . 6 ⊢ (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38		nn0lt10b 12705	. . . . 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 26305	. . . 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 6922	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
45	17	fveq1i 6921	. . . . . . 7 ⊢ (𝑅‘𝐴) = ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴)
46		plyf 26257	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48	47	ffnd 6748	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49		plyf 26257	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
50	7, 49	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
51	50	ffnd 6748	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52		plyf 26257	. . . . . . . . . . . 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 6748	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55		cnex 11265	. . . . . . . . . . 11 ⊢ ℂ ∈ V
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57		inidm 4248	. . . . . . . . . 10 ⊢ (ℂ ∩ ℂ) = ℂ
58	51, 54, 56, 56, 57	offn 7727	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) Fn ℂ)
59		eqidd 2741	. . . . . . . . 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 4201	. . . . . . . . . . . . . . 15 ⊢ (◡𝐺 “ {0}) ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))
62	60, 61	eqsstrrdi 4064	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
63		snssg 4808	. . . . . . . . . . . . . . 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 26341	. . . . . . . . . . . . . 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 2847	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}))
69		fniniseg 7093	. . . . . . . . . . . . 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 7725	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
75	74	anabss3 674	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
76	45, 75	eqtrid 2792	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((𝐹‘𝐴) − 0))
77	46	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) ∈ ℂ)
78	77	subid1d 11636	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹‘𝐴) − 0) = (𝐹‘𝐴))
79	76, 78	eqtrd 2780	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = (𝐹‘𝐴))
80		fvex 6933	. . . . . . 7 ⊢ (𝑅‘0) ∈ V
81	80	fvconst2 7241	. . . . . 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 2788	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝑅‘0))
84	83	sneqd 4660	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹‘𝐴)} = {(𝑅‘0)})
85	84	xpeq2d 5730	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {(𝑅‘0)}))
86	43, 85	eqtr4d 2783	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∪ cun 3974   ⊆ 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  ℂcc 11182  0cc0 11184  1c1 11185   · cmul 11189   < clt 11324   − cmin 11520  ℕ₀cn0 12553  0_𝑝c0p 25723  Polycply 26243  X_pcidp 26244  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-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-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-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:  facth  26366