Theorem rnplynfin 26546

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 26430	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
4	3	frnd 6715	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
5		plyssc 26432	. . . . . . . . . 10 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
6	5, 1	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
8		ssidd 3957	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
9		plyconst 26438	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
10	8, 9	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
11		plysubcl 26455	. . . . . . . 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 2962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (deg‘𝐹) = 0)
15	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ (deg‘𝐹) = 0)
16		0dgr 26478	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (deg‘(ℂ × {𝑥})) = 0)
17	16	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (deg‘(ℂ × {𝑥})) = 0)
18		fveqeq2 6891	. . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
22	21	fconst2 7208	. . . . . . . . . . . 12 ⊢ (𝐹:ℂ⟶{𝑥} ↔ 𝐹 = (ℂ × {𝑥}))
23	20, 22	sylnibr 332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ 𝐹:ℂ⟶{𝑥})
24	3	ffnd 6707	. . . . . . . . . . . . . 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 7215	. . . . . . . . . . . 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 3116	. . . . . . . . . 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 11209	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ ∈ V)
35		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
36		eqidd 2763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) = (𝐹‘𝑦))
37	34, 35, 25, 36	ofc2 7711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
38	37	neeq1d 3016	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 ↔ ((𝐹‘𝑦) − 𝑥) ≠ 0))
39	3	ffvelcdmda 7081	. . . . . . . . . . . . . 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 11605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) = 0 ↔ (𝐹‘𝑦) = 𝑥))
43	42	necon3bid 3001	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) ≠ 0 ↔ (𝐹‘𝑦) ≠ 𝑥))
44		df-ne 2958	. . . . . . . . . . . 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 3186	. . . . . . . . 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 26439	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0) → (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝)
50	49	rexlimiva 3157	. . . . . . . 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 2762	. . . . . . . 8 ⊢ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0})
53	52	fta1 26545	. . . . . . 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 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin)
57		fnconstg 6767	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ × {𝑥}) Fn ℂ)
58	57	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) Fn ℂ)
59		inidm 4175	. . . . . . . . . . . 12 ⊢ (ℂ ∩ ℂ) = ℂ
60	21	fvconst2 7207	. . . . . . . . . . . . 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 7693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
63	62	eqeq1d 2764	. . . . . . . . . 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 7695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 ∘f − (ℂ × {𝑥})) Fn ℂ)
67		fniniseg 7056	. . . . . . . . 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 7056	. . . . . . . . 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 2760	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡𝐹 “ {𝑥}))
73	72	eleq1d 2847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ↔ (◡𝐹 “ {𝑥}) ∈ Fin))
74	73	ralbidva 3185	. . . 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 4003	. . 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 6717	. . . . 5 ⊢ (𝜑 → dom 𝐹 = ℂ)
79		nnnfi 14034	. . . . . . 7 ⊢ ¬ ℕ ∈ Fin
80		nnsscn 12266	. . . . . . . 8 ⊢ ℕ ⊆ ℂ
81		ssfi 9171	. . . . . . . 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 2882	. . . 4 ⊢ (𝜑 → ¬ dom 𝐹 ∈ Fin)
86	85	adantr 486	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) → ¬ dom 𝐹 ∈ Fin)
87	3	ffund 6711	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
88		iunpreima 7065	. . . . . . 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 5023	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ ran 𝐹{𝑥} = ran 𝐹
91	90	imaeq2i 6058	. . . . . . 7 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = (◡𝐹 “ ran 𝐹)
92		cnvimarndm 6083	. . . . . . 7 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
93	91, 92	eqtri 2785	. . . . . 6 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = dom 𝐹
94	89, 93	eqtr3di 2812	. . . . 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 9314	. . . . 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 2863	. . 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 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  {csn 4587  ∪ ciun 4954   class class class wbr 5107   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∘_f cof 7680  Fincfn 8956  ℂcc 11126  0cc0 11128   ≤ cle 11272   − cmin 11469  ℕcn 12261  ♯chash 14398  0_𝑝c0p 25903  Polycply 26416  degcdgr 26419

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 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

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 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-oi 9486  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-rp 13047  df-fz 13566  df-fzo 13714  df-fl 13857  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-rlim 15580  df-sum 15778  df-0p 25904  df-ply 26420  df-idp 26421  df-coe 26422  df-dgr 26423  df-quot 26528

This theorem is used by:  plyconz  26547