plymullem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plymullem1		Structured version Visualization version GIF version

Theorem plymullem1 25719

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 11187	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 15630	. . . 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 15630	. . . 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 7686	. 2 ⊢ (𝜑 → (𝐹 ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) · Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐵‘𝑚) = (𝐵‘𝑛))
11		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑧↑𝑚) = (𝑧↑𝑛))
12	10, 11	oveq12d 7423	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘𝑛) · (𝑧↑𝑛)))
13	12	oveq2d 7421	. . . . . 6 ⊢ (𝑚 = 𝑛 → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))))
14		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝐵‘𝑚) = (𝐵‘(𝑛 − 𝑘)))
15		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑘) → (𝑧↑𝑚) = (𝑧↑(𝑛 − 𝑘)))
16	14, 15	oveq12d 7423	. . . . . . 7 ⊢ (𝑚 = (𝑛 − 𝑘) → ((𝐵‘𝑚) · (𝑧↑𝑚)) = ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘))))
17	16	oveq2d 7421	. . . . . 6 ⊢ (𝑚 = (𝑛 − 𝑘) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑚) · (𝑧↑𝑚))) = (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘(𝑛 − 𝑘)) · (𝑧↑(𝑛 − 𝑘)))))
18		elfznn0 13590	. . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
22		expcl 14041	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
23	22	adantll 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
24	21, 23	mulcld 11230	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
25	18, 24	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
26		elfznn0 13590	. . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ₀) → (𝐵‘𝑛) ∈ ℂ)
30		expcl 14041	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (𝑧↑𝑛) ∈ ℂ)
31	30	adantll 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ₀) → (𝑧↑𝑛) ∈ ℂ)
32	29, 31	mulcld 11230	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ ℕ₀) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
33	26, 32	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
34	25, 33	anim12dan 619	. . . . . . 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 11190	. . . . . . 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 15725	. . . . 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 12530	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℂ)
40	39	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
41		plyaddlem.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
42	41	nn0cnd 12530	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
43	42	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
44		elfznn0 13590	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ₀)
45	44	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ₀)
46	45	nn0cnd 12530	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℂ)
47	40, 43, 46	addsubd 11588	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 − 𝑘) + 𝑁))
48		fznn0sub 13529	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → (𝑀 − 𝑘) ∈ ℕ₀)
49	48	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ ℕ₀)
50		nn0uz 12860	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
51	49, 50	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (𝑀 − 𝑘) ∈ (ℤ_≥‘0))
52	41	nn0zd 12580	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53	52	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
54		eluzadd 12847	. . . . . . . . . . . 12 ⊢ (((𝑀 − 𝑘) ∈ (ℤ_≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ_≥‘(0 + 𝑁)))
55	51, 53, 54	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 − 𝑘) + 𝑁) ∈ (ℤ_≥‘(0 + 𝑁)))
56	47, 55	eqeltrd 2833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ_≥‘(0 + 𝑁)))
57	43	addlidd 11411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0 + 𝑁) = 𝑁)
58	57	fveq2d 6892	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (ℤ_≥‘(0 + 𝑁)) = (ℤ_≥‘𝑁))
59	56, 58	eleqtrd 2835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑀 + 𝑁) − 𝑘) ∈ (ℤ_≥‘𝑁))
60		fzss2 13537	. . . . . . . . 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 593	. . . . . . . . . 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 13590	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℕ₀)
65	64, 32	sylan2 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
66	65	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
67	63, 66	mulcld 11230	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...𝑁)) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
68		eldifn 4126	. . . . . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . . . . . . . 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 12508	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
74	41, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
75	74, 50	eleqtrdi 2843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘0))
76		uzsplit 13569	. . . . . . . . . . . . . . . . . . . . . . 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	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ₀ = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
79		ax-1cn 11164	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
80		pncan 11462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
81	42, 79, 80	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
82	81	oveq2d 7421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
83	82	uneq1d 4161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
84	78, 83	eqtrd 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
85	84	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
86	72, 85	eleqtrd 2835	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → 𝑛 ∈ ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
87		elun 4147	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))) ↔ (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))))
88	86, 87	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑛 ∈ (0...𝑁) ∨ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))))
89	88	ord 862	. . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐵)
92		ssun2 4172	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ_≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1)))
93	92, 78	sseqtrrid 4034	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ ℕ₀)
94	27	fdmd 6725	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐵 = ℕ₀)
95	93, 94	sseqtrrd 4022	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵)
96		funfvima2 7229	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐵 ∧ (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵) → (𝑛 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
97	91, 95, 96	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑛) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
98	97	ad3antrrr 728	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
102	99, 101	eleqtrd 2835	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝐵‘𝑛) ∈ {0})
103		elsni 4644	. . . . . . . . . . . . 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 7420	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = (0 · (𝑧↑𝑛)))
106		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑧 ∈ ℂ)
107	106, 71, 30	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (𝑧↑𝑛) ∈ ℂ)
108	107	mul02d 11408	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑛)) = 0)
109	105, 108	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) = 0)
110	109	oveq2d 7421	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
112	111	mul01d 11409	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ ((0...((𝑀 + 𝑁) − 𝑘)) ∖ (0...𝑁))) → (((𝐴‘𝑘) · (𝑧↑𝑘)) · 0) = 0)
113	110, 112	eqtrd 2772	. . . . . . . 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 13934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
115	61, 67, 113, 114	fsumss 15667	. . . . . . 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 15645	. . . . . 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 13934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ∈ Fin)
118		fzfid 13934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
119	117, 118, 62, 65	fsum2mul 15731	. . . . . 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 11412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
121	41, 50	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
122	38	nn0zd 12580	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
123		eluzadd 12847	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ_≥‘(0 + 𝑀)))
124	121, 122, 123	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ_≥‘(0 + 𝑀)))
125	39	addlidd 11411	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
126	125	fveq2d 6892	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ_≥‘(0 + 𝑀)) = (ℤ_≥‘𝑀))
127	124, 126	eleqtrd 2835	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ_≥‘𝑀))
128	120, 127	eqeltrd 2833	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ_≥‘𝑀))
129		fzss2 13537	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
132	62	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
133	33	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
134	132, 133	mulcld 11230	. . . . . . . 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 15675	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → Σ𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))(((𝐴‘𝑘) · (𝑧↑𝑘)) · ((𝐵‘𝑛) · (𝑧↑𝑛))) ∈ ℂ)
136		eldifn 4126	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
137	136	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
138		eldifi 4125	. . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ℕ₀)
141		peano2nn0 12508	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 + 1) ∈ ℕ₀)
142	38, 141	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
143	142, 50	eleqtrdi 2843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ_≥‘0))
144		uzsplit 13569	. . . . . . . . . . . . . . . . . . . . . . . . 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	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ℕ₀ = ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))))
147		pncan 11462	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
148	39, 79, 147	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
149	148	oveq2d 7421	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
150	149	uneq1d 4161	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
151	146, 150	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
152	151	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
153	140, 152	eleqtrd 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
154		elun 4147	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))) ↔ (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))))
155	153, 154	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))))
156	155	ord 862	. . . . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐴)
159		ssun2 4172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ_≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1)))
160	159, 146	sseqtrrid 4034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ₀)
161	19	fdmd 6725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐴 = ℕ₀)
162	160, 161	sseqtrrd 4022	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴)
163		funfvima2 7229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐴 ∧ (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
164	158, 162, 163	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
165	164	ad2antrr 724	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
169	166, 168	eleqtrd 2835	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
170		elsni 4644	. . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
173	139, 23	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
174	173	mul02d 11408	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
175	172, 174	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
176	175	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
177	176	oveq1d 7420	. . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) ∧ 𝑛 ∈ (0...((𝑀 + 𝑁) − 𝑘))) → ((𝐵‘𝑛) · (𝑧↑𝑛)) ∈ ℂ)
179	178	mul02d 11408	. . . . . . . . . 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 2772	. . . . . . . . 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 15645	. . . . . . . 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 13934	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin)
183	182	olcd 872	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((0...((𝑀 + 𝑁) − 𝑘)) ⊆ (ℤ_≥‘0) ∨ (0...((𝑀 + 𝑁) − 𝑘)) ∈ Fin))
184		sumz 15664	. . . . . . . . 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 2772	. . . . . . 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 13934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
188	131, 135, 186, 187	fsumss 15667	. . . . . 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 2780	. . . . 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 13934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (0...𝑛) ∈ Fin)
191		elfznn0 13590	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑀 + 𝑁)) → 𝑛 ∈ ℕ₀)
192	191, 31	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → (𝑧↑𝑛) ∈ ℂ)
193		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝜑)
194		elfznn0 13590	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ₀)
195	19	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
196	193, 194, 195	syl2an 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴‘𝑘) ∈ ℂ)
197		fznn0sub 13529	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ₀)
198	27	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 𝑘) ∈ ℕ₀) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
199	193, 197, 198	syl2an 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
200	196, 199	mulcld 11230	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
201	190, 192, 200	fsummulc1 15727	. . . . . . 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 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) → 𝑧 ∈ ℂ)
203	202, 194, 22	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑𝑘) ∈ ℂ)
204		expcl 14041	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ (𝑛 − 𝑘) ∈ ℕ₀) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
205	202, 197, 204	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑛 − 𝑘)) ∈ ℂ)
206	196, 203, 199, 205	mul4d 11422	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑧 ∈ ℂ)
208	197	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈ ℕ₀)
209	194	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ₀)
210	207, 208, 209	expaddd 14109	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))))
211	209	nn0cnd 12530	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℂ)
212	191	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℕ₀)
213	212	nn0cnd 12530	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑛 ∈ ℂ)
214	211, 213	pncan3d 11570	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 + (𝑛 − 𝑘)) = 𝑛)
215	214	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧↑(𝑘 + (𝑛 − 𝑘))) = (𝑧↑𝑛))
216	210, 215	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑧↑𝑘) · (𝑧↑(𝑛 − 𝑘))) = (𝑧↑𝑛))
217	216	oveq2d 7421	. . . . . . . . 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 2772	. . . . . . . 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 15645	. . . . . . 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 2775	. . . . . 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 15645	. . . . 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 2783	. . . 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 6888	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
224		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑧↑𝑛) = (𝑧↑𝑘))
225	223, 224	oveq12d 7423	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (𝑧↑𝑛)) = ((𝐵‘𝑘) · (𝑧↑𝑘)))
226	225	cbvsumv 15638	. . . . 5 ⊢ Σ𝑛 ∈ (0...𝑁)((𝐵‘𝑛) · (𝑧↑𝑛)) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))
227	226	oveq2i 7416	. . . 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 2788	. . 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 5247	. 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 2775	1 ⊢ (𝜑 → (𝐹 ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 {csn 4627 ↦ cmpt 5230 dom cdm 5675 “ cima 5678 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 Fincfn 8935 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11440 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ↑cexp 14023 Σcsu 15628 Polycply 25689

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629

This theorem is referenced by: plymullem 25721 coemullem 25755

W3C validator