Theorem plycj 15443

Description: The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on (∗‘𝑧) independently of 𝑧.) (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
plycj.2	⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗)
plycj.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆)
plycj.4	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))

Assertion

Ref	Expression
plycj	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝜑,𝑥

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem plycj

Dummy variables 𝑘 𝑧 𝑎 𝑛 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plycj.4	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
2		elply 15416	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))))
3	1, 2	sylib 122	. . 3 ⊢ (𝜑 → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))))
4	3	simprd 114	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))))
5		simplrl 535	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝑛 ∈ ℕ0)
6		plycj.2	. . . . . . 7 ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗)
7		simplrr 536	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
8		cnex 8131	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
9	8	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
10	3	simpld 112	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
11	9, 10	ssexd 4224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ V)
12	11	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝑆 ∈ V)
13		c0ex 8148	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	snex 4269	. . . . . . . . . 10 ⊢ {0} ∈ V
15		unexg 4534	. . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
16	12, 14, 15	sylancl 413	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → (𝑆 ∪ {0}) ∈ V)
17		nn0ex 9383	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
18	17	a1i 9	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → ℕ0 ∈ V)
19	16, 18	elmapd 6817	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
20	7, 19	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
21		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))))
22		oveq1 6014	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑤↑𝑗) = (𝑧↑𝑗))
23	22	oveq2d 6023	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝑎‘𝑗) · (𝑤↑𝑗)) = ((𝑎‘𝑗) · (𝑧↑𝑗)))
24	23	sumeq2sdv 11889	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
25	24	cbvmptv 4180	. . . . . . . . 9 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
26		fveq2 5629	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑎‘𝑗) = (𝑎‘𝑘))
27		oveq2 6015	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑧↑𝑗) = (𝑧↑𝑘))
28	26, 27	oveq12d 6025	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝑎‘𝑗) · (𝑧↑𝑗)) = ((𝑎‘𝑘) · (𝑧↑𝑘)))
29	28	cbvsumv 11880	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))
30	29	mpteq2i 4171	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
31	25, 30	eqtri 2250	. . . . . . . 8 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
32	21, 31	eqtrdi 2278	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
33	1	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐹 ∈ (Poly‘𝑆))
34	5, 6, 20, 32, 33	plycjlemc 15442	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((∗ ∘ 𝑎)‘𝑘) · (𝑧↑𝑘))))
35		0cn 8146	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
36		snssi 3812	. . . . . . . . . 10 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
38	10, 37	unssd 3380	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
39	38	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → (𝑆 ∪ {0}) ⊆ ℂ)
40	20	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
41		elfznn0 10318	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
42	41	adantl 277	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
43		fvco3 5707	. . . . . . . . 9 ⊢ ((𝑎:ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝑎)‘𝑘) = (∗‘(𝑎‘𝑘)))
44	40, 42, 43	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗ ∘ 𝑎)‘𝑘) = (∗‘(𝑎‘𝑘)))
45	40, 42	ffvelcdmd 5773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ (𝑆 ∪ {0}))
46		plycj.3	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆)
47	46	ralrimiva 2603	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆)
48		fveq2 5629	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎‘𝑘) → (∗‘𝑥) = (∗‘(𝑎‘𝑘)))
49	48	eleq1d 2298	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎‘𝑘) → ((∗‘𝑥) ∈ 𝑆 ↔ (∗‘(𝑎‘𝑘)) ∈ 𝑆))
50	49	rspccv 2904	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆 → ((𝑎‘𝑘) ∈ 𝑆 → (∗‘(𝑎‘𝑘)) ∈ 𝑆))
51	47, 50	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑎‘𝑘) ∈ 𝑆 → (∗‘(𝑎‘𝑘)) ∈ 𝑆))
52		elsni 3684	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘𝑘) ∈ {0} → (𝑎‘𝑘) = 0)
53	52	fveq2d 5633	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = (∗‘0))
54		cj0 11420	. . . . . . . . . . . . . . 15 ⊢ (∗‘0) = 0
55	53, 54	eqtrdi 2278	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = 0)
56	55, 35	eqeltrdi 2320	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ ℂ)
57		elsng 3681	. . . . . . . . . . . . . . 15 ⊢ ((∗‘(𝑎‘𝑘)) ∈ ℂ → ((∗‘(𝑎‘𝑘)) ∈ {0} ↔ (∗‘(𝑎‘𝑘)) = 0))
58	56, 57	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ {0} → ((∗‘(𝑎‘𝑘)) ∈ {0} ↔ (∗‘(𝑎‘𝑘)) = 0))
59	55, 58	mpbird 167	. . . . . . . . . . . . 13 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ {0})
60	59	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ {0}))
61	51, 60	orim12d 791	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}) → ((∗‘(𝑎‘𝑘)) ∈ 𝑆 ∨ (∗‘(𝑎‘𝑘)) ∈ {0})))
62		elun 3345	. . . . . . . . . . 11 ⊢ ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) ↔ ((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}))
63		elun 3345	. . . . . . . . . . 11 ⊢ ((∗‘(𝑎‘𝑘)) ∈ (𝑆 ∪ {0}) ↔ ((∗‘(𝑎‘𝑘)) ∈ 𝑆 ∨ (∗‘(𝑎‘𝑘)) ∈ {0}))
64	61, 62, 63	3imtr4g 205	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) → (∗‘(𝑎‘𝑘)) ∈ (𝑆 ∪ {0})))
65	64	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) → (∗‘(𝑎‘𝑘)) ∈ (𝑆 ∪ {0})))
66	45, 65	mpd 13	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) ∈ (𝑆 ∪ {0}))
67	44, 66	eqeltrd 2306	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗ ∘ 𝑎)‘𝑘) ∈ (𝑆 ∪ {0}))
68	39, 5, 67	elplyd 15423	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((∗ ∘ 𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
69	34, 68	eqeltrd 2306	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐺 ∈ (Poly‘(𝑆 ∪ {0})))
70		plyun0 15418	. . . . 5 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
71	69, 70	eleqtrdi 2322	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝐺 ∈ (Poly‘𝑆))
72	71	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) → (𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) → 𝐺 ∈ (Poly‘𝑆)))
73	72	rexlimdvva 2656	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) → 𝐺 ∈ (Poly‘𝑆)))
74	4, 73	mpd 13	1 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197  {csn 3666   ↦ cmpt 4145   ∘ ccom 4723  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ↑_𝑚 cmap 6803  ℂcc 8005  0cc0 8007   · cmul 8012  ℕ₀cn0 9377  ...cfz 10212  ↑cexp 10768  ∗ccj 11358  Σcsu 11872  Polycply 15410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-fz 10213  df-fzo 10347  df-seqfrec 10678  df-exp 10769  df-ihash 11006  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-clim 11798  df-sumdc 11873  df-ply 15412

This theorem is referenced by: (None)