Theorem plycj 15864

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 15837	. . . 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 541	. . . . . . 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 542	. . . . . . . 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 8303	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
9	8	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
10	3	simpld 112	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
11	9, 10	ssexd 4273	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ V)
12	11	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → 𝑆 ∈ V)
13		c0ex 8320	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	snex 4322	. . . . . . . . . 10 ⊢ {0} ∈ V
15		unexg 4589	. . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
16	12, 14, 15	sylancl 417	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))) ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))) → (𝑆 ∪ {0}) ∈ V)
17		nn0ex 9571	. . . . . . . . . 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 6936	. . . . . . . 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 6092	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑤↑𝑗) = (𝑧↑𝑗))
23	22	oveq2d 6101	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝑎‘𝑗) · (𝑤↑𝑗)) = ((𝑎‘𝑗) · (𝑧↑𝑗)))
24	23	sumeq2sdv 12138	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
25	24	cbvmptv 4227	. . . . . . . . 9 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)))
26		fveq2 5695	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑎‘𝑗) = (𝑎‘𝑘))
27		oveq2 6093	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑧↑𝑗) = (𝑧↑𝑘))
28	26, 27	oveq12d 6103	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝑎‘𝑗) · (𝑧↑𝑗)) = ((𝑎‘𝑘) · (𝑧↑𝑘)))
29	28	cbvsumv 12129	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))
30	29	mpteq2i 4218	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
31	25, 30	eqtri 2259	. . . . . . . 8 ⊢ (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
32	21, 31	eqtrdi 2287	. . . . . . 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 492	. . . . . . 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 15863	. . . . . 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 8318	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
36		snssi 3859	. . . . . . . . . 10 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
38	10, 37	unssd 3405	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
39	38	ad2antrr 492	. . . . . . 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 10523	. . . . . . . . . 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 5776	. . . . . . . . 9 ⊢ ((𝑎:ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝑎)‘𝑘) = (∗‘(𝑎‘𝑘)))
44	40, 42, 43	syl2anc 415	. . . . . . . 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 5844	. . . . . . . . 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 2623	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆)
48		fveq2 5695	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎‘𝑘) → (∗‘𝑥) = (∗‘(𝑎‘𝑘)))
49	48	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎‘𝑘) → ((∗‘𝑥) ∈ 𝑆 ↔ (∗‘(𝑎‘𝑘)) ∈ 𝑆))
50	49	rspccv 2926	. . . . . . . . . . . . 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 3727	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘𝑘) ∈ {0} → (𝑎‘𝑘) = 0)
53	52	fveq2d 5699	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = (∗‘0))
54		cj0 11669	. . . . . . . . . . . . . . 15 ⊢ (∗‘0) = 0
55	53, 54	eqtrdi 2287	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) = 0)
56	55, 35	eqeltrdi 2329	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘𝑘) ∈ {0} → (∗‘(𝑎‘𝑘)) ∈ ℂ)
57		elsng 3724	. . . . . . . . . . . . . . 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 798	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}) → ((∗‘(𝑎‘𝑘)) ∈ 𝑆 ∨ (∗‘(𝑎‘𝑘)) ∈ {0})))
62		elun 3370	. . . . . . . . . . 11 ⊢ ((𝑎‘𝑘) ∈ (𝑆 ∪ {0}) ↔ ((𝑎‘𝑘) ∈ 𝑆 ∨ (𝑎‘𝑘) ∈ {0}))
63		elun 3370	. . . . . . . . . . 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 496	. . . . . . . . 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 2315	. . . . . . 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 15844	. . . . . 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 2315	. . . . 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 15839	. . . . 5 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
71	69, 70	eleqtrdi 2331	. . . 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 2676	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3709   ↦ cmpt 4192   ∘ ccom 4778  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ↑_𝑚 cmap 6922  ℂcc 8177  0cc0 8179   · cmul 8184  ℕ₀cn0 9565  ...cfz 10413  ↑cexp 10977  ∗ccj 11606  Σcsu 12121  Polycply 15831

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-seqfrec 10887  df-exp 10978  df-ihash 11217  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-clim 12047  df-sumdc 12122  df-ply 15833

This theorem is used by: (None)