Theorem plyrem 24060

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15260). 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 23956	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 473	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sseldi 3601	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4		plyrem.1	. . . . . . . . . 10 ⊢ 𝐺 = (Xp ∘𝑓 − (ℂ × {𝐴}))
5	4	plyremlem 24059	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
6	5	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
7	6	simp1d 1073	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
8	6	simp2d 1074	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9		ax-1ne0 10005	. . . . . . . . . 10 ⊢ 1 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
11	8, 10	eqnetrd 2861	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12		fveq2 6191	. . . . . . . . . 10 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13		dgr0 24018	. . . . . . . . . 10 ⊢ (deg‘0𝑝) = 0
14	12, 13	syl6eq 2672	. . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
15	14	necon3i 2826	. . . . . . . 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 24058	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
19	3, 7, 16, 18	syl3anc 1326	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20		0lt1 10550	. . . . . . . 8 ⊢ 0 < 1
21	20, 8	syl5breqr 4691	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22		fveq2 6191	. . . . . . . . 9 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
23	22, 13	syl6eq 2672	. . . . . . . 8 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
24	23	breq1d 4663	. . . . . . 7 ⊢ (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
25	21, 24	syl5ibrcom 237	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26		pm2.62 425	. . . . . 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 4679	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29		quotcl2 24057	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
30	3, 7, 16, 29	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31		plymulcl 23977	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
32	7, 30, 31	syl2anc 693	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33		plysubcl 23978	. . . . . . . 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 693	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
35	17, 34	syl5eqel 2705	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36		dgrcl 23989	. . . . . 6 ⊢ (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38		nn0lt10b 11439	. . . . 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 222	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41		0dgrb 24002	. . . 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 222	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
44	43	fveq1d 6193	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
45	17	fveq1i 6192	. . . . . . 7 ⊢ (𝑅‘𝐴) = ((𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺)))‘𝐴)
46		plyf 23954	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48		ffn 6045	. . . . . . . . . 10 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
50		plyf 23954	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
51	7, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
52		ffn 6045	. . . . . . . . . . 11 ⊢ (𝐺:ℂ⟶ℂ → 𝐺 Fn ℂ)
53	51, 52	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
54		plyf 23954	. . . . . . . . . . . 12 ⊢ ((𝐹 quot 𝐺) ∈ (Poly‘ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
55	30, 54	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
56		ffn 6045	. . . . . . . . . . 11 ⊢ ((𝐹 quot 𝐺):ℂ⟶ℂ → (𝐹 quot 𝐺) Fn ℂ)
57	55, 56	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
58		cnex 10017	. . . . . . . . . . 11 ⊢ ℂ ∈ V
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
60		inidm 3822	. . . . . . . . . 10 ⊢ (ℂ ∩ ℂ) = ℂ
61	53, 57, 59, 59, 60	offn 6908	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)) Fn ℂ)
62		eqidd 2623	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝐹‘𝐴))
63	6	simp3d 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡𝐺 “ {0}) = {𝐴})
64		ssun1 3776	. . . . . . . . . . . . . . 15 ⊢ (◡𝐺 “ {0}) ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))
65	63, 64	syl6eqssr 3656	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
66		snssg 4327	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
67	66	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
68	65, 67	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
69		ofmulrt 24037	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∈ V ∧ 𝐺:ℂ⟶ℂ ∧ (𝐹 quot 𝐺):ℂ⟶ℂ) → (◡(𝐺 ∘𝑓 · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
70	59, 51, 55, 69	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡(𝐺 ∘𝑓 · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
71	68, 70	eleqtrrd 2704	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ (◡(𝐺 ∘𝑓 · (𝐹 quot 𝐺)) “ {0}))
72		fniniseg 6338	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∘𝑓 · (𝐹 quot 𝐺)) Fn ℂ → (𝐴 ∈ (◡(𝐺 ∘𝑓 · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺 ∘𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)))
73	61, 72	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (◡(𝐺 ∘𝑓 · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺 ∘𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)))
74	71, 73	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺 ∘𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0))
75	74	simprd 479	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
76	75	adantr 481	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
77	49, 61, 59, 59, 60, 62, 76	ofval 6906	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
78	77	anabss3 864	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
79	45, 78	syl5eq 2668	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((𝐹‘𝐴) − 0))
80	46	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) ∈ ℂ)
81	80	subid1d 10381	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹‘𝐴) − 0) = (𝐹‘𝐴))
82	79, 81	eqtrd 2656	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = (𝐹‘𝐴))
83		fvex 6201	. . . . . . 7 ⊢ (𝑅‘0) ∈ V
84	83	fvconst2 6469	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
85	84	adantl 482	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
86	44, 82, 85	3eqtr3d 2664	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝑅‘0))
87	86	sneqd 4189	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹‘𝐴)} = {(𝑅‘0)})
88	87	xpeq2d 5139	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {(𝑅‘0)}))
89	43, 88	eqtr4d 2659	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℂcc 9934  0cc0 9936  1c1 9937   · cmul 9941   < clt 10074   − cmin 10266  ℕ₀cn0 11292  0_𝑝c0p 23436  Polycply 23940  X_pcidp 23941  degcdgr 23943   quot cquot 24045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  facth  24061