Theorem rnplynfin 26631

Description: The range of a nonconstant polynomial is not a finite set. (Contributed by SN, 30-Aug-2026.)

Hypotheses

Ref	Expression
rnplynfin.f	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
rnplynfin.1	⊢ (𝜑 → (deg‘𝐹) ≠ 0)

Assertion

Ref	Expression
rnplynfin	⊢ (𝜑 → ¬ ran 𝐹 ∈ Fin)

Proof of Theorem rnplynfin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnplynfin.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
2		plyf 26517	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
4	3	frnd 6718	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
5		plyssc 26519	. . . . . . . . . 10 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
6	5, 1	sselid 3906	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
8		ssidd 3931	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
9		plyconst 26525	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
10	8, 9	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
11		plysubcl 26542	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (ℂ × {𝑥}) ∈ (Poly‘ℂ)) → (𝐹 ∘f − (ℂ × {𝑥})) ∈ (Poly‘ℂ))
12	7, 10, 11	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 ∘f − (ℂ × {𝑥})) ∈ (Poly‘ℂ))
13		rnplynfin.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (deg‘𝐹) ≠ 0)
14	13	neneqd 2960	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (deg‘𝐹) = 0)
15	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ (deg‘𝐹) = 0)
16		0dgr 26565	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (deg‘(ℂ × {𝑥})) = 0)
17	16	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (deg‘(ℂ × {𝑥})) = 0)
18		fveqeq2 6894	. . . . . . . . . . . . . 14 ⊢ (𝐹 = (ℂ × {𝑥}) → ((deg‘𝐹) = 0 ↔ (deg‘(ℂ × {𝑥})) = 0))
19	17, 18	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 = (ℂ × {𝑥}) → (deg‘𝐹) = 0))
20	15, 19	mtod 201	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ 𝐹 = (ℂ × {𝑥}))
21		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
22	21	fconst2 7211	. . . . . . . . . . . 12 ⊢ (𝐹:ℂ⟶{𝑥} ↔ 𝐹 = (ℂ × {𝑥}))
23	20, 22	sylnibr 332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ 𝐹:ℂ⟶{𝑥})
24	3	ffnd 6710	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn ℂ)
25	24	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐹 Fn ℂ)
26	25	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥) → 𝐹 Fn ℂ)
27		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥) → ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥)
28		fconstfv 7218	. . . . . . . . . . . 12 ⊢ (𝐹:ℂ⟶{𝑥} ↔ (𝐹 Fn ℂ ∧ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥))
29	26, 27, 28	sylanbrc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥) → 𝐹:ℂ⟶{𝑥})
30	23, 29	mtand 828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥)
31		rexnal 3114	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℂ ¬ (𝐹‘𝑦) = 𝑥 ↔ ¬ ∀𝑦 ∈ ℂ (𝐹‘𝑦) = 𝑥)
32	30, 31	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∃𝑦 ∈ ℂ ¬ (𝐹‘𝑦) = 𝑥)
33		cnex 11282	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ ∈ V)
35		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
36		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) = (𝐹‘𝑦))
37	34, 35, 25, 36	ofc2 7722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
38	37	neeq1d 3014	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 ↔ ((𝐹‘𝑦) − 𝑥) ≠ 0))
39	3	ffvelcdmda 7084	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) ∈ ℂ)
40	39	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) ∈ ℂ)
41		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
42	40, 41	subeq0ad 11678	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) = 0 ↔ (𝐹‘𝑦) = 𝑥))
43	42	necon3bid 2999	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) ≠ 0 ↔ (𝐹‘𝑦) ≠ 𝑥))
44		df-ne 2956	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ≠ 𝑥 ↔ ¬ (𝐹‘𝑦) = 𝑥)
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹‘𝑦) ≠ 𝑥 ↔ ¬ (𝐹‘𝑦) = 𝑥))
46	38, 43, 45	3bitrd 308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 ↔ ¬ (𝐹‘𝑦) = 𝑥))
47	46	rexbidva 3184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (∃𝑦 ∈ ℂ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 ↔ ∃𝑦 ∈ ℂ ¬ (𝐹‘𝑦) = 𝑥))
48	32, 47	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∃𝑦 ∈ ℂ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0)
49		ne0p 26526	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0) → (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝)
50	49	rexlimiva 3155	. . . . . . . 8 ⊢ (∃𝑦 ∈ ℂ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 → (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝)
51	48, 50	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝)
52		eqid 2760	. . . . . . . 8 ⊢ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0})
53	52	fta1 26630	. . . . . . 7 ⊢ (((𝐹 ∘f − (ℂ × {𝑥})) ∈ (Poly‘ℂ) ∧ (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝) → ((◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0})) ≤ (deg‘(𝐹 ∘f − (ℂ × {𝑥})))))
54	12, 51, 53	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0})) ≤ (deg‘(𝐹 ∘f − (ℂ × {𝑥})))))
55	54	simpld 500	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin)
56	55	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin)
57		fnconstg 6770	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ × {𝑥}) Fn ℂ)
58	57	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) Fn ℂ)
59		inidm 4123	. . . . . . . . . . . 12 ⊢ (ℂ ∩ ℂ) = ℂ
60	21	fvconst2 7210	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → ((ℂ × {𝑥})‘𝑦) = 𝑥)
61	60	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((ℂ × {𝑥})‘𝑦) = 𝑥)
62	25, 58, 34, 34, 59, 36, 61	ofval 7704	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
63	62	eqeq1d 2762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = 0 ↔ ((𝐹‘𝑦) − 𝑥) = 0))
64	63, 42	bitrd 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = 0 ↔ (𝐹‘𝑦) = 𝑥))
65	64	pm5.32da 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = 0) ↔ (𝑦 ∈ ℂ ∧ (𝐹‘𝑦) = 𝑥)))
66	25, 58, 34, 34, 59	offn 7706	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 ∘f − (ℂ × {𝑥})) Fn ℂ)
67		fniniseg 7059	. . . . . . . . 9 ⊢ ((𝐹 ∘f − (ℂ × {𝑥})) Fn ℂ → (𝑦 ∈ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ↔ (𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = 0)))
68	66, 67	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑦 ∈ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ↔ (𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = 0)))
69		fniniseg 7059	. . . . . . . . 9 ⊢ (𝐹 Fn ℂ → (𝑦 ∈ (◡𝐹 “ {𝑥}) ↔ (𝑦 ∈ ℂ ∧ (𝐹‘𝑦) = 𝑥)))
70	25, 69	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑦 ∈ (◡𝐹 “ {𝑥}) ↔ (𝑦 ∈ ℂ ∧ (𝐹‘𝑦) = 𝑥)))
71	65, 68, 70	3bitr4d 314	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑦 ∈ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ↔ 𝑦 ∈ (◡𝐹 “ {𝑥})))
72	71	eqrdv 2758	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡𝐹 “ {𝑥}))
73	72	eleq1d 2845	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ↔ (◡𝐹 “ {𝑥}) ∈ Fin))
74	73	ralbidva 3183	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ℂ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ↔ ∀𝑥 ∈ ℂ (◡𝐹 “ {𝑥}) ∈ Fin))
75	56, 74	mpbid 235	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (◡𝐹 “ {𝑥}) ∈ Fin)
76		ssralv 3977	. . 3 ⊢ (ran 𝐹 ⊆ ℂ → (∀𝑥 ∈ ℂ (◡𝐹 “ {𝑥}) ∈ Fin → ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin))
77	4, 75, 76	sylc 66	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin)
78	3	fdmd 6720	. . . . 5 ⊢ (𝜑 → dom 𝐹 = ℂ)
79		nnnfi 14110	. . . . . . 7 ⊢ ¬ ℕ ∈ Fin
80		nnsscn 12341	. . . . . . . 8 ⊢ ℕ ⊆ ℂ
81		ssfi 9189	. . . . . . . 8 ⊢ ((ℂ ∈ Fin ∧ ℕ ⊆ ℂ) → ℕ ∈ Fin)
82	80, 81	mpan2 704	. . . . . . 7 ⊢ (ℂ ∈ Fin → ℕ ∈ Fin)
83	79, 82	mto 200	. . . . . 6 ⊢ ¬ ℂ ∈ Fin
84	83	a1i 11	. . . . 5 ⊢ (𝜑 → ¬ ℂ ∈ Fin)
85	78, 84	eqneltrd 2880	. . . 4 ⊢ (𝜑 → ¬ dom 𝐹 ∈ Fin)
86	85	adantr 486	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) → ¬ dom 𝐹 ∈ Fin)
87	3	ffund 6714	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
88		iunpreima 7068	. . . . . . 7 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
89	87, 88	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
90		iunid 5018	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ ran 𝐹{𝑥} = ran 𝐹
91	90	imaeq2i 6049	. . . . . . 7 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = (◡𝐹 “ ran 𝐹)
92		cnvimarndm 6079	. . . . . . 7 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
93	91, 92	eqtri 2783	. . . . . 6 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = dom 𝐹
94	89, 93	eqtr3di 2810	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) = dom 𝐹)
95	94	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) ∧ ran 𝐹 ∈ Fin) → ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) = dom 𝐹)
96		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
97		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) ∧ ran 𝐹 ∈ Fin) → ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin)
98		iunfi 9333	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) → ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin)
99	96, 97, 98	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) ∧ ran 𝐹 ∈ Fin) → ∪ 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin)
100	95, 99	eqeltrrd 2861	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin)
101	86, 100	mtand 828	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) → ¬ ran 𝐹 ∈ Fin)
102	77, 101	mpdan 700	1 ⊢ (𝜑 → ¬ ran 𝐹 ∈ Fin)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  {csn 4583  ∪ ciun 4950   class class class wbr 5102   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691  Fincfn 8974  ℂcc 11199  0cc0 11201   ≤ cle 11345   − cmin 11542  ℕcn 12336  ♯chash 14475  0_𝑝c0p 25991  Polycply 26503  degcdgr 26506

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278  ax-pre-sup 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-oadd 8481  df-er 8718  df-map 8850  df-pm 8851  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9435  df-inf 9436  df-oi 9505  df-dju 9983  df-card 10021  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-div 11975  df-nn 12337  df-2 12406  df-3 12407  df-n0 12608  df-xnn0 12681  df-z 12695  df-uz 12967  df-rp 13122  df-fz 13641  df-fzo 13790  df-fl 13933  df-seq 14146  df-exp 14206  df-hash 14476  df-cj 15267  df-re 15268  df-im 15269  df-sqrt 15403  df-abs 15404  df-clim 15656  df-rlim 15657  df-sum 15855  df-0p 25992  df-ply 26507  df-idp 26508  df-coe 26509  df-dgr 26510  df-quot 26613

This theorem is used by:  plyconz  26632