Theorem plymullem1 24915

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 10661	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 15097	. . . 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 15097	. . . 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 7429	. 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fveq2 6662	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐵‘𝑚) = (𝐵‘𝑛))
11		oveq2 7163	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑧↑𝑚) = (𝑧↑𝑛))
12	10, 11	oveq12d 7173	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘𝑛) · (𝑧↑𝑛)))
13	12	oveq2d 7171	. . . . . 6 ⊢ (𝑚 = 𝑛 → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
14		fveq2 6662	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝐵‘𝑚) = (𝐵‘(𝑛 − 𝑘)))
15		oveq2 7163	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝑧↑𝑚) = (𝑧↑(𝑛 − 𝑘)))
16	14, 15	oveq12d 7173	. . . . . . 7 ⊢ (𝑚 = (𝑛 − 𝑘) → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘))))
17	16	oveq2d 7171	. . . . . 6 ⊢ (𝑚 = (𝑛 − 𝑘) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
18		elfznn0 13054	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
19		plyaddlem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
20	19	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ)
21	20	ffvelrnda 6847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
22		expcl 13502	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
23	22	adantll 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
24	21, 23	mulcld 10704	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
25	18, 24	sylan2 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
26		elfznn0 13054	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘)) → 𝑛 ∈ ℕ0)
27		plyaddlem.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
28	27	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ0⟶ℂ)
29	28	ffvelrnda 6847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → (𝐵‘𝑛) ∈ ℂ)
30		expcl 13502	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) ∈ ℂ)
31	30	adantll 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) ∈ ℂ)
32	29, 31	mulcld 10704	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ0) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
33	26, 32	sylan2 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
34	25, 33	anim12dan 621	. . . . . . 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 10664	. . . . . . 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 15191	. . . . 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 12001	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℂ)
40	39	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
41		plyaddlem.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
42	41	nn0cnd 12001	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
43	42	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
44		elfznn0 13054	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ0)
45	44	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ0)
46	45	nn0cnd 12001	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℂ)
47	40, 43, 46	addsubd 11061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 − 𝑘) + 𝑁))
48		fznn0sub 12993	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → (𝑀 − 𝑘) ∈ ℕ0)
49	48	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ ℕ0)
50		nn0uz 12325	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
51	49, 50	eleqtrdi 2862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ (ℤ≥‘0))
52	41	nn0zd 12129	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53	52	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
54		eluzadd 12318	. . . . . . . . . . . 12 ⊢ (((𝑀 − 𝑘) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
55	51, 53, 54	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
56	47, 55	eqeltrd 2852	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ≥‘(0 + 𝑁)))
57	43	addid2d 10884	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0 + 𝑁) = 𝑁)
58	57	fveq2d 6666	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (ℤ≥‘(0 + 𝑁)) = (ℤ≥‘𝑁))
59	56, 58	eleqtrd 2854	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ≥‘𝑁))
60		fzss2 13001	. . . . . . . . 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 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
63	62	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
64		elfznn0 13054	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℕ0)
65	64, 32	sylan2 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
66	65	adantlr 714	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
67	63, 66	mulcld 10704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
68		eldifn 4035	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁)) → ¬ 𝑛 ∈ (0...𝑁))
69	68	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ¬ 𝑛 ∈ (0...𝑁))
70		eldifi 4034	. . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ ℕ0)
73		peano2nn0 11979	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
74	41, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
75	74, 50	eleqtrdi 2862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘0))
76		uzsplit 13033	. . . . . . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))))
79		ax-1cn 10638	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
80		pncan 10935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
81	42, 79, 80	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
82	81	oveq2d 7171	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
83	82	uneq1d 4069	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
84	78, 83	eqtrd 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
85	84	ad3antrrr 729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
86	72, 85	eleqtrd 2854	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
87		elun 4056	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))) ↔ (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))))
88	86, 87	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))))
89	88	ord 861	. . . . . . . . . . . . . . . 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	27	ffund 6506	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐵)
92		ssun2 4080	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))
93	92, 78	sseqtrrid 3947	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
94	27	fdmd 6512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐵 = ℕ0)
95	93, 94	sseqtrrd 3935	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐵)
96		funfvima2 6990	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐵 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐵) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
97	91, 95, 96	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
98	97	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1)))))
99	90, 98	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ≥‘(𝑁 + 1))))
100		plyaddlem.b2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
101	100	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
102	99, 101	eleqtrd 2854	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) ∈ {0})
103		elsni 4542	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑛) ∈ {0} → (𝐵‘𝑛) = 0)
104	102, 103	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) = 0)
105	104	oveq1d 7170	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = (0 · (𝑧↑𝑛)))
106		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑧 ∈ ℂ)
107	106, 71, 30	syl2an 598	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑧↑𝑛) ∈ ℂ)
108	107	mul02d 10881	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑛)) = 0)
109	105, 108	eqtrd 2793	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = 0)
110	109	oveq2d 7171	. . . . . . . . 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))
111	62	adantr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
112	111	mul01d 10882	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · 0) = 0)
113	110, 112	eqtrd 2793	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
114		fzfid 13395	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
115	61, 67, 113, 114	fsumss 15135	. . . . . . 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))))
116	115	sumeq2dv 15113	. . . . . 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))))
117		fzfid 13395	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ∈ Fin)
118		fzfid 13395	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
119	117, 118, 62, 65	fsum2mul 15197	. . . . . 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))))
120	39, 42	addcomd 10885	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
121	41, 50	eleqtrdi 2862	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
122	38	nn0zd 12129	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
123		eluzadd 12318	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
124	121, 122, 123	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
125	39	addid2d 10884	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
126	125	fveq2d 6666	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑀)) = (ℤ≥‘𝑀))
127	124, 126	eleqtrd 2854	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘𝑀))
128	120, 127	eqeltrd 2852	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑀))
129		fzss2 13001	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
130	128, 129	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
131	130	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
132	62	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
133	33	adantlr 714	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
134	132, 133	mulcld 10704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
135	114, 134	fsumcl 15143	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
136		eldifn 4035	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
137	136	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
138		eldifi 4034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
139	138, 18	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ ℕ0)
140	139	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ℕ0)
141		peano2nn0 11979	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
142	38, 141	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ0)
143	142, 50	eleqtrdi 2862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘0))
144		uzsplit 13033	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
145	143, 144	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (ℤ≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
146	50, 145	syl5eq 2805	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ℕ0 = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
147		pncan 10935	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
148	39, 79, 147	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
149	148	oveq2d 7171	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
150	149	uneq1d 4069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
151	146, 150	eqtrd 2793	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ℕ0 = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
152	151	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ℕ0 = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
153	140, 152	eleqtrd 2854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
154		elun 4056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
155	153, 154	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
156	155	ord 861	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (¬ 𝑘 ∈ (0...𝑀) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1))))
157	137, 156	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
158	19	ffund 6506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐴)
159		ssun2 4080	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1)))
160	159, 146	sseqtrrid 3947	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ0)
161	19	fdmd 6512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐴 = ℕ0)
162	160, 161	sseqtrrd 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ dom 𝐴)
163		funfvima2 6990	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑀 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
164	158, 162, 163	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
165	164	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑘 ∈ (ℤ≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1)))))
166	157, 165	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑀 + 1))))
167		plyaddlem.a2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
168	167	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
169	166, 168	eleqtrd 2854	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
170		elsni 4542	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑘) ∈ {0} → (𝐴‘𝑘) = 0)
171	169, 170	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) = 0)
172	171	oveq1d 7170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
173	139, 23	sylan2 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
174	173	mul02d 10881	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
175	172, 174	eqtrd 2793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
176	175	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
177	176	oveq1d 7170	. . . . . . . . . 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))))
178	33	adantlr 714	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
179	178	mul02d 10881	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (0 · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
180	177, 179	eqtrd 2793	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
181	180	sumeq2dv 15113	. . . . . . . 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)
182		fzfid 13395	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
183	182	olcd 871	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((0...((𝑀 + 𝑁) − 𝑘)) ⊆ (ℤ≥‘0) ∨ (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin))
184		sumz 15132	. . . . . . . . 9 ⊢ (((0...((𝑀 + 𝑁) − 𝑘)) ⊆ (ℤ≥‘0) ∨ (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))0 = 0)
185	183, 184	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))0 = 0)
186	181, 185	eqtrd 2793	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) = 0)
187		fzfid 13395	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
188	131, 135, 186, 187	fsumss 15135	. . . . . 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))))
189	116, 119, 188	3eqtr3d 2801	. . . . 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))))
190		fzfid 13395	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (0...𝑛) ∈ Fin)
191		elfznn0 13054	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑀 + 𝑁)) → 𝑛 ∈ ℕ0)
192	191, 31	sylan2 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (𝑧↑𝑛) ∈ ℂ)
193		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝜑)
194		elfznn0 13054	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
195	19	ffvelrnda 6847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
196	193, 194, 195	syl2an 598	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴‘𝑘) ∈ ℂ)
197		fznn0sub 12993	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
198	27	ffvelrnda 6847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 𝑘) ∈ ℕ0) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
199	193, 197, 198	syl2an 598	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
200	196, 199	mulcld 10704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
201	190, 192, 200	fsummulc1 15193	. . . . . . 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)))
202		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝑧 ∈ ℂ)
203	202, 194, 22	syl2an 598	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑𝑘) ∈ ℂ)
204		expcl 13502	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ (𝑛 − 𝑘) ∈ ℕ0) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
205	202, 197, 204	syl2an 598	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
206	196, 203, 199, 205	mul4d 10895	. . . . . . . . 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)))))
207	202	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑧 ∈ ℂ)
208	197	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈ ℕ0)
209	194	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
210	207, 208, 209	expaddd 13567	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))))
211	209	nn0cnd 12001	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℂ)
212	191	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℕ0)
213	212	nn0cnd 12001	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℂ)
214	211, 213	pncan3d 11043	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 + (𝑛 − 𝑘)) = 𝑛)
215	214	oveq2d 7171	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = (𝑧↑𝑛))
216	210, 215	eqtr3d 2795	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))) = (𝑧↑𝑛))
217	216	oveq2d 7171	. . . . . . . . 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)))
218	206, 217	eqtrd 2793	. . . . . . . 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)))
219	218	sumeq2dv 15113	. . . . . . 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)))
220	201, 219	eqtr4d 2796	. . . . . 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)))))
221	220	sumeq2dv 15113	. . . . 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)))))
222	37, 189, 221	3eqtr4rd 2804	. . . 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))))
223		fveq2 6662	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
224		oveq2 7163	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑧↑𝑛) = (𝑧↑𝑘))
225	223, 224	oveq12d 7173	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (𝑧↑𝑛)) = ((𝐵‘𝑘) · (𝑧↑𝑘)))
226	225	cbvsumv 15106	. . . . 5 ⊢ Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛)) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))
227	226	oveq2i 7166	. . . 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)))
228	222, 227	eqtrdi 2809	. . 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))))
229	228	mpteq2dva 5130	. 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)))))
230	9, 229	eqtr4d 2796	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   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  {csn 4525   ↦ cmpt 5115  dom cdm 5527   “ cima 5530  Fun wfun 6333  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∘_f cof 7408  Fincfn 8532  ℂcc 10578  0cc0 10580  1c1 10581   + caddc 10583   · cmul 10585   − cmin 10913  ℕ₀cn0 11939  ℤcz 12025  ℤ_≥cuz 12287  ...cfz 12944  ↑cexp 13484  Σcsu 15095  Polycply 24885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-sup 8944  df-oi 9012  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-3 11743  df-n0 11940  df-z 12026  df-uz 12288  df-rp 12436  df-fz 12945  df-fzo 13088  df-seq 13424  df-exp 13485  df-hash 13746  df-cj 14511  df-re 14512  df-im 14513  df-sqrt 14647  df-abs 14648  df-clim 14898  df-sum 15096

This theorem is referenced by:  plymullem  24917  coemullem  24951