Theorem rnplynfin 26660

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 26546	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
4	3	frnd 6740	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
5		plyssc 26548	. . . . . . . . . 10 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
6	5, 1	sselid 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
8		ssidd 3936	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
9		plyconst 26554	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
10	8, 9	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) ∈ (Poly‘ℂ))
11		plysubcl 26571	. . . . . . . 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 2964	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (deg‘𝐹) = 0)
15	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ (deg‘𝐹) = 0)
16		0dgr 26594	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (deg‘(ℂ × {𝑥})) = 0)
17	16	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (deg‘(ℂ × {𝑥})) = 0)
18		fveqeq2 6917	. . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
22	21	fconst2 7234	. . . . . . . . . . . 12 ⊢ (𝐹:ℂ⟶{𝑥} ↔ 𝐹 = (ℂ × {𝑥}))
23	20, 22	sylnibr 332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ¬ 𝐹:ℂ⟶{𝑥})
24	3	ffnd 6732	. . . . . . . . . . . . . 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 7241	. . . . . . . . . . . 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 3118	. . . . . . . . . 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 11308	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ ∈ V)
35		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
36		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) = (𝐹‘𝑦))
37	34, 35, 25, 36	ofc2 7746	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
38	37	neeq1d 3018	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0 ↔ ((𝐹‘𝑦) − 𝑥) ≠ 0))
39	3	ffvelcdmda 7108	. . . . . . . . . . . . . 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 11704	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) = 0 ↔ (𝐹‘𝑦) = 𝑥))
43	42	necon3bid 3003	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (((𝐹‘𝑦) − 𝑥) ≠ 0 ↔ (𝐹‘𝑦) ≠ 𝑥))
44		df-ne 2960	. . . . . . . . . . . 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 3188	. . . . . . . . 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 26555	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) ≠ 0) → (𝐹 ∘f − (ℂ × {𝑥})) ≠ 0𝑝)
50	49	rexlimiva 3159	. . . . . . . 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 2763	. . . . . . . 8 ⊢ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0})
53	52	fta1 26659	. . . . . . 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 3158	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin)
57		fnconstg 6792	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ × {𝑥}) Fn ℂ)
58	57	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ × {𝑥}) Fn ℂ)
59		inidm 4126	. . . . . . . . . . . 12 ⊢ (ℂ ∩ ℂ) = ℂ
60	21	fvconst2 7233	. . . . . . . . . . . . 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 7728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝐹 ∘f − (ℂ × {𝑥}))‘𝑦) = ((𝐹‘𝑦) − 𝑥))
63	62	eqeq1d 2765	. . . . . . . . . 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 7730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 ∘f − (ℂ × {𝑥})) Fn ℂ)
67		fniniseg 7083	. . . . . . . . 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 7083	. . . . . . . . 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 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) = (◡𝐹 “ {𝑥}))
73	72	eleq1d 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((◡(𝐹 ∘f − (ℂ × {𝑥})) “ {0}) ∈ Fin ↔ (◡𝐹 “ {𝑥}) ∈ Fin))
74	73	ralbidva 3187	. . . 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 3982	. . 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 6742	. . . . 5 ⊢ (𝜑 → dom 𝐹 = ℂ)
79		nnnfi 14136	. . . . . . 7 ⊢ ¬ ℕ ∈ Fin
80		nnsscn 12367	. . . . . . . 8 ⊢ ℕ ⊆ ℂ
81		ssfi 9216	. . . . . . . 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 2884	. . . 4 ⊢ (𝜑 → ¬ dom 𝐹 ∈ Fin)
86	85	adantr 486	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}) ∈ Fin) → ¬ dom 𝐹 ∈ Fin)
87	3	ffund 6736	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
88		iunpreima 7092	. . . . . . 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 5022	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ ran 𝐹{𝑥} = ran 𝐹
91	90	imaeq2i 6062	. . . . . . 7 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = (◡𝐹 “ ran 𝐹)
92		cnvimarndm 6092	. . . . . . 7 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
93	91, 92	eqtri 2786	. . . . . 6 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ ran 𝐹{𝑥}) = dom 𝐹
94	89, 93	eqtr3di 2813	. . . . 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 9360	. . . . 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 2865	. . 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  {csn 4588  ∪ ciun 4954   class class class wbr 5106   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  Fun wfun 6552   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715  Fincfn 9001  ℂcc 11225  0cc0 11227   ≤ cle 11371   − cmin 11568  ℕcn 12362  ♯chash 14501  0_𝑝c0p 26020  Polycply 26532  degcdgr 26535

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-inf 9462  df-oi 9531  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-rp 13148  df-fz 13667  df-fzo 13816  df-fl 13959  df-seq 14172  df-exp 14232  df-hash 14502  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-clim 15684  df-rlim 15685  df-sum 15883  df-0p 26021  df-ply 26536  df-idp 26537  df-coe 26538  df-dgr 26539  df-quot 26642

This theorem is used by:  plyconz  26661