Theorem plyrem 26536

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16637). 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 26427	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 488	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3932	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4		plyrem.1	. . . . . . . . . 10 ⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))
5	4	plyremlem 26535	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
6	5	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
7	6	simp1d 1160	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
8	6	simp2d 1161	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9		ax-1ne0 11196	. . . . . . . . . 10 ⊢ 1 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
11	8, 10	eqnetrd 3024	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12		fveq2 6882	. . . . . . . . . 10 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13		dgr0 26489	. . . . . . . . . 10 ⊢ (deg‘0𝑝) = 0
14	12, 13	eqtrdi 2813	. . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
15	14	necon3i 2989	. . . . . . . 8 ⊢ ((deg‘𝐺) ≠ 0 → 𝐺 ≠ 0𝑝)
16	11, 15	syl 18	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ≠ 0𝑝)
17		plyrem.2	. . . . . . . 8 ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))
18	17	quotdgr 26534	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
19	3, 7, 16, 18	syl3anc 1398	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20		0lt1 11763	. . . . . . . 8 ⊢ 0 < 1
21	20, 8	breqtrrid 5147	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22		fveq2 6882	. . . . . . . . 9 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
23	22, 13	eqtrdi 2813	. . . . . . . 8 ⊢ (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
24	23	breq1d 5117	. . . . . . 7 ⊢ (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
25	21, 24	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26		pm2.62 913	. . . . . 6 ⊢ ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) → ((𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)) → (deg‘𝑅) < (deg‘𝐺)))
27	19, 25, 26	sylc 66	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < (deg‘𝐺))
28	27, 8	breqtrd 5135	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29		quotcl2 26533	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
30	3, 7, 16, 29	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31		plymulcl 26448	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
32	7, 30, 31	syl2anc 596	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33		plysubcl 26449	. . . . . . . 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 596	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
35	17, 34	eqeltrid 2866	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36		dgrcl 26460	. . . . . 6 ⊢ (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
37	35, 36	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38		nn0lt10b 12686	. . . . 5 ⊢ ((deg‘𝑅) ∈ ℕ0 → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
39	37, 38	syl 18	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
40	28, 39	mpbid 235	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41		0dgrb 26473	. . . 4 ⊢ (𝑅 ∈ (Poly‘ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
42	35, 41	syl 18	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
43	40, 42	mpbid 235	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
44	43	fveq1d 6884	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
45	17	fveq1i 6883	. . . . . . 7 ⊢ (𝑅‘𝐴) = ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴)
46		plyf 26425	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47	46	adantr 486	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48	47	ffnd 6707	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49		plyf 26425	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
50	7, 49	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
51	50	ffnd 6707	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52		plyf 26425	. . . . . . . . . . . 12 ⊢ ((𝐹 quot 𝐺) ∈ (Poly‘ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
53	30, 52	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
54	53	ffnd 6707	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55		cnex 11208	. . . . . . . . . . 11 ⊢ ℂ ∈ V
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57		inidm 4175	. . . . . . . . . 10 ⊢ (ℂ ∩ ℂ) = ℂ
58	51, 54, 56, 56, 57	offn 7694	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∘f · (𝐹 quot 𝐺)) Fn ℂ)
59		eqidd 2763	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝐹‘𝐴))
60	6	simp3d 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡𝐺 “ {0}) = {𝐴})
61		ssun1 4127	. . . . . . . . . . . . . . 15 ⊢ (◡𝐺 “ {0}) ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))
62	60, 61	eqsstrrdi 3979	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
63		snssg 4747	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
64	63	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0}))))
65	62, 64	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
66		ofmulrt 26510	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}) = ((◡𝐺 “ {0}) ∪ (◡(𝐹 quot 𝐺) “ {0})))
68	65, 67	eleqtrrd 2865	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ (◡(𝐺 ∘f · (𝐹 quot 𝐺)) “ {0}))
69		fniniseg 7056	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0))
72	71	simprd 501	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
73	72	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺 ∘f · (𝐹 quot 𝐺))‘𝐴) = 0)
74	48, 58, 56, 56, 57, 59, 73	ofval 7692	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
75	74	anabss3 688	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹‘𝐴) − 0))
76	45, 75	eqtrid 2809	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = ((𝐹‘𝐴) − 0))
77	46	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) ∈ ℂ)
78	77	subid1d 11585	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹‘𝐴) − 0) = (𝐹‘𝐴))
79	76, 78	eqtrd 2797	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅‘𝐴) = (𝐹‘𝐴))
80		fvex 6895	. . . . . . 7 ⊢ (𝑅‘0) ∈ V
81	80	fvconst2 7206	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
82	81	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
83	44, 79, 82	3eqtr3d 2805	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = (𝑅‘0))
84	83	sneqd 4599	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹‘𝐴)} = {(𝑅‘0)})
85	84	xpeq2d 5689	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {(𝑅‘0)}))
86	43, 85	eqtr4d 2800	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  {csn 4587   class class class wbr 5107   × cxp 5657  ^◡ccnv 5658   “ cima 5662   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∘_f cof 7679  ℂcc 11125  0cc0 11127  1c1 11128   · cmul 11132   < clt 11270   − cmin 11468  ℕ₀cn0 12531  0_𝑝c0p 25898  Polycply 26411  X_pcidp 26412  degcdgr 26414   quot cquot 26521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-pm 8832  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13045  df-fz 13564  df-fzo 13712  df-fl 13855  df-seq 14068  df-exp 14128  df-hash 14397  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-clim 15577  df-rlim 15578  df-sum 15776  df-0p 25899  df-ply 26415  df-idp 26416  df-coe 26417  df-dgr 26418  df-quot 26522

This theorem is used by:  facth  26537