Theorem plycj 15081

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 15054	. . . 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 8020	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
9	8	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
10	3	simpld 112	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
11	9, 10	ssexd 4174	. . . . . . . . . . 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 8037	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	snex 4219	. . . . . . . . . 10 ⊢ {0} ∈ V
15		unexg 4479	. . . . . . . . . 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 9272	. . . . . . . . . 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 6730	. . . . . . . 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 5932	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑤↑𝑗) = (𝑧↑𝑗))
23	22	oveq2d 5941	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝑎‘𝑗) · (𝑤↑𝑗)) = ((𝑎‘𝑗) · (𝑧↑𝑗)))
24	23	sumeq2sdv 11552	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
25	24	cbvmptv 4130	. . . . . . . . 9 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
26		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑎‘𝑗) = (𝑎‘𝑘))
27		oveq2 5933	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑧↑𝑗) = (𝑧↑𝑘))
28	26, 27	oveq12d 5943	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝑎‘𝑗) · (𝑧↑𝑗)) = ((𝑎‘𝑘) · (𝑧↑𝑘)))
29	28	cbvsumv 11543	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))
30	29	mpteq2i 4121	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
31	25, 30	eqtri 2217	. . . . . . . 8 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
32	21, 31	eqtrdi 2245	. . . . . . 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 15080	. . . . . 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 8035	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
36		snssi 3767	. . . . . . . . . 10 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
38	10, 37	unssd 3340	. . . . . . . 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 10206	. . . . . . . . . 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 5635	. . . . . . . . 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 5701	. . . . . . . . 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 2570	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆)
48		fveq2 5561	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎‘𝑘) → (∗‘𝑥) = (∗‘(𝑎‘𝑘)))
49	48	eleq1d 2265	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎‘𝑘) → ((∗‘𝑥) ∈ 𝑆 ↔ (∗‘(𝑎‘𝑘)) ∈ 𝑆))
50	49	rspccv 2865	. . . . . . . . . . . . 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 3641	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘𝑘) ∈ {0} → (𝑎‘𝑘) = 0)
53	52	fveq2d 5565	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = (∗‘0))
54		cj0 11083	. . . . . . . . . . . . . . 15 ⊢ (∗‘0) = 0
55	53, 54	eqtrdi 2245	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = 0)
56	55, 35	eqeltrdi 2287	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ ℂ)
57		elsng 3638	. . . . . . . . . . . . . . 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 787	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}) → ((∗‘(𝑎‘𝑘)) ∈ 𝑆 ∨ (∗‘(𝑎‘𝑘)) ∈ {0})))
62		elun 3305	. . . . . . . . . . 11 ⊢ ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) ↔ ((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}))
63		elun 3305	. . . . . . . . . . 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 2273	. . . . . . 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 15061	. . . . . 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 2273	. . . . 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 15056	. . . . 5 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
71	69, 70	eleqtrdi 2289	. . . 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 2622	. 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 709   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  {csn 3623   ↦ cmpt 4095   ∘ ccom 4668  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925   ↑_𝑚 cmap 6716  ℂcc 7894  0cc0 7896   · cmul 7901  ℕ₀cn0 9266  ...cfz 10100  ↑cexp 10647  ∗ccj 11021  Σcsu 11535  Polycply 15048

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ply 15050

This theorem is referenced by: (None)