Theorem plymullem1 23970

Description: Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)

Hypotheses

Ref	Expression
plyaddlem.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
plyaddlem.2	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
plyaddlem.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
plyaddlem.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
plyaddlem.a	⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
plyaddlem.b	⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
plyaddlem.a2	⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
plyaddlem.b2	⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
plyaddlem.f	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
plyaddlem.g	⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))

Assertion

Ref	Expression
plymullem1	⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))))

Distinct variable groups:   𝐴,𝑛   𝑘,𝑛,𝐵   𝑘,𝑀,𝑛   𝑘,𝑁,𝑛   𝑧,𝑘,𝜑,𝑛

Allowed substitution hints:   𝐴(𝑧,𝑘)   𝐵(𝑧)   𝑆(𝑧,𝑘,𝑛)   𝐹(𝑧,𝑘,𝑛)   𝐺(𝑧,𝑘,𝑛)   𝑀(𝑧)   𝑁(𝑧)

Proof of Theorem plymullem1

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 10017	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 14418	. . . 4 ⊢ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V)
5		sumex 14418	. . . 4 ⊢ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V)
7		plyaddlem.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
8		plyaddlem.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
9	2, 4, 6, 7, 8	offval2 6914	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐵‘𝑚) = (𝐵‘𝑛))
11		oveq2 6658	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑧↑𝑚) = (𝑧↑𝑛))
12	10, 11	oveq12d 6668	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘𝑛) · (𝑧↑𝑛)))
13	12	oveq2d 6666	. . . . . 6 ⊢ (𝑚 = 𝑛 → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
14		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝐵‘𝑚) = (𝐵‘(𝑛 − 𝑘)))
15		oveq2 6658	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝑧↑𝑚) = (𝑧↑(𝑛 − 𝑘)))
16	14, 15	oveq12d 6668	. . . . . . 7 ⊢ (𝑚 = (𝑛 − 𝑘) → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘))))
17	16	oveq2d 6666	. . . . . 6 ⊢ (𝑚 = (𝑛 − 𝑘) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
18		elfznn0 12433	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
19		plyaddlem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
20	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ)
21	20	ffvelrnda 6359	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
22		expcl 12878	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
23	22	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
24	21, 23	mulcld 10060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
25	18, 24	sylan2 491	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
26		elfznn0 12433	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘)) → 𝑛 ∈ ℕ0)
27		plyaddlem.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ0⟶ℂ)
29	28	ffvelrnda 6359	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → (𝐵‘𝑛) ∈ ℂ)
30		expcl 12878	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) ∈ ℂ)
31	30	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) ∈ ℂ)
32	29, 31	mulcld 10060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
33	26, 32	sylan2 491	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
34	25, 33	anim12dan 882	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘)))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ ∧ ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ))
35		mulcl 10020	. . . . . . 7 ⊢ ((((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ ∧ ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
36	34, 35	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘)))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
37	13, 17, 36	fsum0diag2 14515	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑛 ∈ (0...(𝑀 + 𝑁))Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
38		plyaddlem.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
39	38	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℂ)
40	39	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
41		plyaddlem.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
42	41	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
43	42	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
44		elfznn0 12433	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ0)
45	44	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ0)
46	45	nn0cnd 11353	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℂ)
47	40, 43, 46	addsubd 10413	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 − 𝑘) + 𝑁))
48		fznn0sub 12373	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → (𝑀 − 𝑘) ∈ ℕ0)
49	48	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ ℕ0)
50		nn0uz 11722	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
51	49, 50	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ (ℤ≥‘0))
52	41	nn0zd 11480	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53	52	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
54		eluzadd 11716	. . . . . . . . . . . 12 ⊢ (((𝑀 − 𝑘) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
55	51, 53, 54	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
56	47, 55	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ≥‘(0 + 𝑁)))
57	43	addid2d 10237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0 + 𝑁) = 𝑁)
58	57	fveq2d 6195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (ℤ≥‘(0 + 𝑁)) = (ℤ≥‘𝑁))
59	56, 58	eleqtrd 2703	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ≥‘𝑁))
60		fzss2 12381	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 𝑘) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...((𝑀 + 𝑁) − 𝑘)))
61	59, 60	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0...𝑁) ⊆ (0...((𝑀 + 𝑁) − 𝑘)))
62	44, 24	sylan2 491	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
63	62	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
64		elfznn0 12433	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℕ0)
65	64, 32	sylan2 491	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
66	65	adantlr 751	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
67	63, 66	mulcld 10060	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
68		eldifn 3733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁)) → ¬ 𝑛 ∈ (0...𝑁))
69	68	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ¬ 𝑛 ∈ (0...𝑁))
70		eldifi 3732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁)) → 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘)))
71	70, 26	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁)) → 𝑛 ∈ ℕ0)
72	71	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ ℕ0)
73		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
74	41, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
75	74, 50	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘0))
76		uzsplit 12412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))))
77	75, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (ℤ≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))))
78	50, 77	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))))
79		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
80		pncan 10287	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
81	42, 79, 80	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
82	81	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
83	82	uneq1d 3766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
84	78, 83	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
85	84	ad3antrrr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
86	72, 85	eleqtrd 2703	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
87		elun 3753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))) ↔ (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))))
88	86, 87	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))))
89	88	ord 392	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (¬ 𝑛 ∈ (0...𝑁) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1))))
90	69, 89	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
91		ffun 6048	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵:ℕ0⟶ℂ → Fun 𝐵)
92	27, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐵)
93		ssun2 3777	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))
94	93, 78	syl5sseqr 3654	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
95		fdm 6051	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵:ℕ0⟶ℂ → dom 𝐵 = ℕ0)
96	27, 95	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐵 = ℕ0)
97	94, 96	sseqtr4d 3642	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐵)
98		funfvima2 6493	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐵 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐵) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
99	92, 97, 98	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
100	99	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
101	90, 100	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1))))
102		plyaddlem.b2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
103	102	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
104	101, 103	eleqtrd 2703	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) ∈ {0})
105		elsni 4194	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑛) ∈ {0} → (𝐵‘𝑛) = 0)
106	104, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) = 0)
107	106	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = (0 · (𝑧↑𝑛)))
108		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑧 ∈ ℂ)
109	108, 71, 30	syl2an 494	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑧↑𝑛) ∈ ℂ)
110	109	mul02d 10234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑛)) = 0)
111	107, 110	eqtrd 2656	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = 0)
112	111	oveq2d 6666	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · 0))
113	62	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
114	113	mul01d 10235	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · 0) = 0)
115	112, 114	eqtrd 2656	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
116		fzfid 12772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
117	61, 67, 115, 116	fsumss 14456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → Σ𝑛 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
118	117	sumeq2dv 14433	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)Σ𝑛 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑘 ∈ (0...𝑀)Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
119		fzfid 12772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ∈ Fin)
120		fzfid 12772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
121	119, 120, 62, 65	fsum2mul 14521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)Σ𝑛 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛))))
122	39, 42	addcomd 10238	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
123	41, 50	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
124	38	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
125		eluzadd 11716	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
126	123, 124, 125	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
127	39	addid2d 10237	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
128	127	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑀)) = (ℤ≥‘𝑀))
129	126, 128	eleqtrd 2703	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘𝑀))
130	122, 129	eqeltrd 2701	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑀))
131		fzss2 12381	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
132	130, 131	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
133	132	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
134	62	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
135	33	adantlr 751	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
136	134, 135	mulcld 10060	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
137	116, 136	fsumcl 14464	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
138		eldifn 3733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
139	138	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
140		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
141	140, 18	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ ℕ0)
142	141	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ℕ0)
143		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
144	38, 143	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ0)
145	144, 50	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘0))
146		uzsplit 12412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
147	145, 146	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (ℤ≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
148	50, 147	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ℕ0 = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
149		pncan 10287	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
150	39, 79, 149	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
151	150	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
152	151	uneq1d 3766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
153	148, 152	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ℕ0 = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
154	153	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ℕ0 = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
155	142, 154	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
156		elun 3753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
157	155, 156	sylib 208	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
158	157	ord 392	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (¬ 𝑘 ∈ (0...𝑀) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
159	139, 158	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
160		ffun 6048	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴:ℕ0⟶ℂ → Fun 𝐴)
161	19, 160	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐴)
162		ssun2 3777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1)))
163	162, 148	syl5sseqr 3654	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ0)
164		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴:ℕ0⟶ℂ → dom 𝐴 = ℕ0)
165	19, 164	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐴 = ℕ0)
166	163, 165	sseqtr4d 3642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ dom 𝐴)
167		funfvima2 6493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑀 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
168	161, 166, 167	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
169	168	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
170	159, 169	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1))))
171		plyaddlem.a2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
172	171	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
173	170, 172	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
174		elsni 4194	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑘) ∈ {0} → (𝐴‘𝑘) = 0)
175	173, 174	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) = 0)
176	175	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
177	141, 23	sylan2 491	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
178	177	mul02d 10234	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
179	176, 178	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
180	179	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
181	180	oveq1d 6665	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = (0 · ((𝐵‘𝑛) · (𝑧↑𝑛))))
182	33	adantlr 751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
183	182	mul02d 10234	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (0 · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
184	181, 183	eqtrd 2656	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
185	184	sumeq2dv 14433	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))0)
186		fzfid 12772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
187	186	olcd 408	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((0...((𝑀 + 𝑁) − 𝑘)) ⊆ (ℤ≥‘0) ∨ (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin))
188		sumz 14453	. . . . . . . . 9 ⊢ (((0...((𝑀 + 𝑁) − 𝑘)) ⊆ (ℤ≥‘0) ∨ (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))0 = 0)
189	187, 188	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))0 = 0)
190	185, 189	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
191		fzfid 12772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
192	133, 137, 190, 191	fsumss 14456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
193	118, 121, 192	3eqtr3d 2664	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛))) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
194		fzfid 12772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (0...𝑛) ∈ Fin)
195		elfznn0 12433	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑀 + 𝑁)) → 𝑛 ∈ ℕ0)
196	195, 31	sylan2 491	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (𝑧↑𝑛) ∈ ℂ)
197		simpll 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝜑)
198		elfznn0 12433	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
199	19	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
200	197, 198, 199	syl2an 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴‘𝑘) ∈ ℂ)
201		fznn0sub 12373	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
202	27	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 𝑘) ∈ ℕ0) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
203	197, 201, 202	syl2an 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
204	200, 203	mulcld 10060	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
205	194, 196, 204	fsummulc1 14517	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)) = Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)))
206		simplr 792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝑧 ∈ ℂ)
207	206, 198, 22	syl2an 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑𝑘) ∈ ℂ)
208		expcl 12878	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ (𝑛 − 𝑘) ∈ ℕ0) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
209	206, 201, 208	syl2an 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
210	200, 207, 203, 209	mul4d 10248	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))) = (((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘)))))
211	206	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑧 ∈ ℂ)
212	201	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈ ℕ0)
213	198	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
214	211, 212, 213	expaddd 13010	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))))
215	213	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℂ)
216	195	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℕ0)
217	216	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℂ)
218	215, 217	pncan3d 10395	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 + (𝑛 − 𝑘)) = 𝑛)
219	218	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = (𝑧↑𝑛))
220	214, 219	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))) = (𝑧↑𝑛))
221	220	oveq2d 6666	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘)))) = (((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)))
222	210, 221	eqtrd 2656	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))) = (((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)))
223	222	sumeq2dv 14433	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))) = Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)))
224	205, 223	eqtr4d 2659	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)) = Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
225	224	sumeq2dv 14433	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)) = Σ𝑛 ∈ (0...(𝑀 + 𝑁))Σ𝑘 ∈ (0...𝑛)(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
226	37, 193, 225	3eqtr4rd 2667	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)) = (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛))))
227		fveq2 6191	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
228		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑧↑𝑛) = (𝑧↑𝑘))
229	227, 228	oveq12d 6668	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (𝑧↑𝑛)) = ((𝐵‘𝑘) · (𝑧↑𝑘)))
230	229	cbvsumv 14426	. . . . 5 ⊢ Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛)) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))
231	230	oveq2i 6661	. . . 4 ⊢ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛))) = (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
232	226, 231	syl6eq 2672	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)) = (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
233	232	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
234	9, 233	eqtr4d 2659	1 ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  {csn 4177   ↦ cmpt 4729  dom cdm 5114   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  Fincfn 7955  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  Σcsu 14416  Polycply 23940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  plymullem  23972  coemullem  24006