Theorem plycj 15675

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 15648	. . . 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 537	. . . . . . 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 538	. . . . . . . 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 8256	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
9	8	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
10	3	simpld 112	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
11	9, 10	ssexd 4252	. . . . . . . . . . 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 8273	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	snex 4300	. . . . . . . . . 10 ⊢ {0} ∈ V
15		unexg 4566	. . . . . . . . . 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 9507	. . . . . . . . . 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 6898	. . . . . . . 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 6059	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑤↑𝑗) = (𝑧↑𝑗))
23	22	oveq2d 6068	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝑎‘𝑗) · (𝑤↑𝑗)) = ((𝑎‘𝑗) · (𝑧↑𝑗)))
24	23	sumeq2sdv 12063	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
25	24	cbvmptv 4208	. . . . . . . . 9 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
26		fveq2 5672	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑎‘𝑗) = (𝑎‘𝑘))
27		oveq2 6060	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑧↑𝑗) = (𝑧↑𝑘))
28	26, 27	oveq12d 6070	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝑎‘𝑗) · (𝑧↑𝑗)) = ((𝑎‘𝑘) · (𝑧↑𝑘)))
29	28	cbvsumv 12054	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))
30	29	mpteq2i 4199	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
31	25, 30	eqtri 2255	. . . . . . . 8 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
32	21, 31	eqtrdi 2283	. . . . . . 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 15674	. . . . . 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 8271	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
36		snssi 3840	. . . . . . . . . 10 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
38	10, 37	unssd 3397	. . . . . . . 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 10455	. . . . . . . . . 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 5750	. . . . . . . . 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 5815	. . . . . . . . 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 2617	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆)
48		fveq2 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎‘𝑘) → (∗‘𝑥) = (∗‘(𝑎‘𝑘)))
49	48	eleq1d 2303	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎‘𝑘) → ((∗‘𝑥) ∈ 𝑆 ↔ (∗‘(𝑎‘𝑘)) ∈ 𝑆))
50	49	rspccv 2920	. . . . . . . . . . . . 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 3709	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘𝑘) ∈ {0} → (𝑎‘𝑘) = 0)
53	52	fveq2d 5676	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = (∗‘0))
54		cj0 11594	. . . . . . . . . . . . . . 15 ⊢ (∗‘0) = 0
55	53, 54	eqtrdi 2283	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = 0)
56	55, 35	eqeltrdi 2325	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ ℂ)
57		elsng 3706	. . . . . . . . . . . . . . 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 794	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}) → ((∗‘(𝑎‘𝑘)) ∈ 𝑆 ∨ (∗‘(𝑎‘𝑘)) ∈ {0})))
62		elun 3362	. . . . . . . . . . 11 ⊢ ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) ↔ ((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}))
63		elun 3362	. . . . . . . . . . 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 2311	. . . . . . 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 15655	. . . . . 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 2311	. . . . 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 15650	. . . . 5 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
71	69, 70	eleqtrdi 2327	. . . 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 2670	. 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 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ∪ cun 3211   ⊆ wss 3213  {csn 3691   ↦ cmpt 4173   ∘ ccom 4755  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052   ↑_𝑚 cmap 6884  ℂcc 8130  0cc0 8132   · cmul 8137  ℕ₀cn0 9501  ...cfz 10348  ↑cexp 10907  ∗ccj 11532  Σcsu 12046  Polycply 15642

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-seqfrec 10817  df-exp 10908  df-ihash 11147  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-sumdc 12047  df-ply 15644

This theorem is referenced by: (None)