Theorem breprexplemc 30710

Description: Lemma for breprexp 30711 (induction step) (Contributed by Thierry Arnoux, 6-Dec-2021.)

Hypotheses

Ref	Expression
breprexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
breprexp.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
breprexp.z	⊢ (𝜑 → 𝑍 ∈ ℂ)
breprexp.h	⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
breprexplemc.t	⊢ (𝜑 → 𝑇 ∈ ℕ0)
breprexplemc.s	⊢ (𝜑 → (𝑇 + 1) ≤ 𝑆)
breprexplemc.1	⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))

Assertion

Ref	Expression
breprexplemc	⊢ (𝜑 → ∏𝑎 ∈ (0..^(𝑇 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))

Distinct variable groups:   𝑚,𝑁   𝑆,𝑎,𝑚   𝑚,𝑍   𝑚,𝐿   𝑇,𝑎,𝑏,𝑑,𝑚   𝑍,𝑎,𝑏,𝑑   𝐿,𝑎,𝑏,𝑑   𝜑,𝑎,𝑏,𝑑,𝑚   𝑁,𝑎,𝑏,𝑑

Allowed substitution hints:   𝑆(𝑏,𝑑)

Proof of Theorem breprexplemc

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breprexplemc.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℕ0)
2		nn0uz 11722	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	syl6eleq 2711	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘0))
4		fzosplitsn 12576	. . . 4 ⊢ (𝑇 ∈ (ℤ≥‘0) → (0..^(𝑇 + 1)) = ((0..^𝑇) ∪ {𝑇}))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (0..^(𝑇 + 1)) = ((0..^𝑇) ∪ {𝑇}))
6	5	prodeq1d 14651	. 2 ⊢ (𝜑 → ∏𝑎 ∈ (0..^(𝑇 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ ((0..^𝑇) ∪ {𝑇})Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)))
7		nfv 1843	. . 3 ⊢ Ⅎ𝑎𝜑
8		nfcv 2764	. . 3 ⊢ Ⅎ𝑎Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))
9		fzofi 12773	. . . 4 ⊢ (0..^𝑇) ∈ Fin
10	9	a1i 11	. . 3 ⊢ (𝜑 → (0..^𝑇) ∈ Fin)
11		fzonel 12483	. . . 4 ⊢ ¬ 𝑇 ∈ (0..^𝑇)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (0..^𝑇))
13		fzfid 12772	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → (1...𝑁) ∈ Fin)
14		breprexp.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
15	14	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑁 ∈ ℕ0)
16		breprexp.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
17	16	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
18		breprexp.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℂ)
19	18	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
20		breprexp.h	. . . . . . . 8 ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
21	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
22	21	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
23	1	nn0zd 11480	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℤ)
24	16	nn0zd 11480	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℤ)
25	1	nn0red 11352	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℝ)
26		1red 10055	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
27	25, 26	readdcld 10069	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 + 1) ∈ ℝ)
28	16	nn0red 11352	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℝ)
29	25	lep1d 10955	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ≤ (𝑇 + 1))
30		breprexplemc.s	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 + 1) ≤ 𝑆)
31	25, 27, 28, 29, 30	letrd 10194	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≤ 𝑆)
32		eluz1 11691	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ℤ → (𝑆 ∈ (ℤ≥‘𝑇) ↔ (𝑆 ∈ ℤ ∧ 𝑇 ≤ 𝑆)))
33	32	biimpar 502	. . . . . . . . . 10 ⊢ ((𝑇 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑇 ≤ 𝑆)) → 𝑆 ∈ (ℤ≥‘𝑇))
34	23, 24, 31, 33	syl12anc 1324	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘𝑇))
35		fzoss2 12496	. . . . . . . . 9 ⊢ (𝑆 ∈ (ℤ≥‘𝑇) → (0..^𝑇) ⊆ (0..^𝑆))
36	34, 35	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑇) ⊆ (0..^𝑆))
37	36	sselda 3603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑆))
38	37	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑎 ∈ (0..^𝑆))
39		fz1ssnn 12372	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → (1...𝑁) ⊆ ℕ)
41	40	sselda 3603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ)
42	15, 17, 19, 22, 38, 41	breprexplemb 30709	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑎)‘𝑏) ∈ ℂ)
43		nnssnn0 11295	. . . . . . . . . . 11 ⊢ ℕ ⊆ ℕ0
44	39, 43	sstri 3612	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ0
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ0)
46	45	ralrimivw 2967	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (0..^𝑇)(1...𝑁) ⊆ ℕ0)
47	46	r19.21bi 2932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → (1...𝑁) ⊆ ℕ0)
48	47	sselda 3603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
49	19, 48	expcld 13008	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
50	42, 49	mulcld 10060	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
51	13, 50	fsumcl 14464	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑇)) → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
52		simpl 473	. . . . . . 7 ⊢ ((𝑎 = 𝑇 ∧ 𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑇)
53	52	fveq2d 6195	. . . . . 6 ⊢ ((𝑎 = 𝑇 ∧ 𝑏 ∈ (1...𝑁)) → (𝐿‘𝑎) = (𝐿‘𝑇))
54	53	fveq1d 6193	. . . . 5 ⊢ ((𝑎 = 𝑇 ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑎)‘𝑏) = ((𝐿‘𝑇)‘𝑏))
55	54	oveq1d 6665	. . . 4 ⊢ ((𝑎 = 𝑇 ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)))
56	55	sumeq2dv 14433	. . 3 ⊢ (𝑎 = 𝑇 → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)))
57		fzfid 12772	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
58	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑁 ∈ ℕ0)
59	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
60	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
61	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
62	1	nn0ge0d 11354	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑇)
63		zltp1le 11427	. . . . . . . . . 10 ⊢ ((𝑇 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑇 < 𝑆 ↔ (𝑇 + 1) ≤ 𝑆))
64	23, 24, 63	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 < 𝑆 ↔ (𝑇 + 1) ≤ 𝑆))
65	30, 64	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → 𝑇 < 𝑆)
66		0zd 11389	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℤ)
67		elfzo 12472	. . . . . . . . 9 ⊢ ((𝑇 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑇 ∈ (0..^𝑆) ↔ (0 ≤ 𝑇 ∧ 𝑇 < 𝑆)))
68	23, 66, 24, 67	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∈ (0..^𝑆) ↔ (0 ≤ 𝑇 ∧ 𝑇 < 𝑆)))
69	62, 65, 68	mpbir2and 957	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (0..^𝑆))
70	69	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑇 ∈ (0..^𝑆))
71	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
72	71	sselda 3603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ)
73	58, 59, 60, 61, 70, 72	breprexplemb 30709	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
74	45	sselda 3603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
75	60, 74	expcld 13008	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
76	73, 75	mulcld 10060	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
77	57, 76	fsumcl 14464	. . 3 ⊢ (𝜑 → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
78	7, 8, 10, 1, 12, 51, 56, 77	fprodsplitsn 14720	. 2 ⊢ (𝜑 → ∏𝑎 ∈ ((0..^𝑇) ∪ {𝑇})Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) · Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
79		breprexplemc.1	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))
80	79	oveq1d 6665	. . 3 ⊢ (𝜑 → (∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) · Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = (Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
81		fzfid 12772	. . . 4 ⊢ (𝜑 → (0...(𝑇 · 𝑁)) ∈ Fin)
82	39	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → (1...𝑁) ⊆ ℕ)
83		fz0ssnn0 12435	. . . . . . . 8 ⊢ (0...(𝑇 · 𝑁)) ⊆ ℕ0
84		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → 𝑚 ∈ (0...(𝑇 · 𝑁)))
85	83, 84	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → 𝑚 ∈ ℕ0)
86	85	nn0zd 11480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → 𝑚 ∈ ℤ)
87	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → 𝑇 ∈ ℕ0)
88	57	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → (1...𝑁) ∈ Fin)
89	82, 86, 87, 88	reprfi 30694	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → ((1...𝑁)(repr‘𝑇)𝑚) ∈ Fin)
90	9	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (0..^𝑇) ∈ Fin)
91	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → 𝑁 ∈ ℕ0)
92	91	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑁 ∈ ℕ0)
93	16	ad3antrrr 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑆 ∈ ℕ0)
94	18	ad3antrrr 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑍 ∈ ℂ)
95	20	ad3antrrr 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
96	36	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (0..^𝑇) ⊆ (0..^𝑆))
97	96	sselda 3603	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑆))
98	39	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → (1...𝑁) ⊆ ℕ)
99	86	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑚 ∈ ℤ)
100	87	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑇 ∈ ℕ0)
101		simplr 792	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚))
102	98, 99, 100, 101	reprf 30690	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑑:(0..^𝑇)⟶(1...𝑁))
103		simpr 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑇))
104	102, 103	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ (1...𝑁))
105	39, 104	sseldi 3601	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ ℕ)
106	92, 93, 94, 95, 97, 105	breprexplemb 30709	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
107	90, 106	fprodcl 14682	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
108	18	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑍 ∈ ℂ)
109	85	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑚 ∈ ℕ0)
110	108, 109	expcld 13008	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (𝑍↑𝑚) ∈ ℂ)
111	107, 110	mulcld 10060	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) ∈ ℂ)
112	89, 111	fsumcl 14464	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) ∈ ℂ)
113	81, 57, 112, 76	fsum2mul 14521	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = (Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
114	39	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
115		fzssz 12343	. . . . . . . . . . . . 13 ⊢ (0...((𝑇 + 1) · 𝑁)) ⊆ ℤ
116		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑚 ∈ (0...((𝑇 + 1) · 𝑁)))
117	115, 116	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑚 ∈ ℤ)
118	117	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑚 ∈ ℤ)
119		fzssz 12343	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℤ
120		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
121	119, 120	sseldi 3601	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
122	118, 121	zsubcld 11487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (𝑚 − 𝑏) ∈ ℤ)
123	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑇 ∈ ℕ0)
124	123	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑇 ∈ ℕ0)
125	57	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (1...𝑁) ∈ Fin)
126	125	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
127	114, 122, 124, 126	reprfi 30694	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏)) ∈ Fin)
128	73	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
129	18	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑍 ∈ ℂ)
130		fz0ssnn0 12435	. . . . . . . . . . . . 13 ⊢ (0...((𝑇 + 1) · 𝑁)) ⊆ ℕ0
131	130, 116	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑚 ∈ ℕ0)
132	129, 131	expcld 13008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (𝑍↑𝑚) ∈ ℂ)
133	132	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑚) ∈ ℂ)
134	128, 133	mulcld 10060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)) ∈ ℂ)
135	9	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → (0..^𝑇) ∈ Fin)
136	14	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑁 ∈ ℕ0)
137	136	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑁 ∈ ℕ0)
138	137	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑁 ∈ ℕ0)
139	16	ad4antr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑆 ∈ ℕ0)
140	129	ad3antrrr 766	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑍 ∈ ℂ)
141	20	ad4antr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
142	37	ad5ant15 1303	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑆))
143	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → (1...𝑁) ⊆ ℕ)
144	122	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑚 − 𝑏) ∈ ℤ)
145	124	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑇 ∈ ℕ0)
146		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏)))
147	143, 144, 145, 146	reprf 30690	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑑:(0..^𝑇)⟶(1...𝑁))
148		simpr 477	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑇))
149	147, 148	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ (1...𝑁))
150	39, 149	sseldi 3601	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ ℕ)
151	138, 139, 140, 141, 142, 150	breprexplemb 30709	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑇)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
152	135, 151	fprodcl 14682	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
153	127, 134, 152	fsummulc1 14517	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
154	153	sumeq2dv 14433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
155		elfzle2 12345	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...((𝑇 + 1) · 𝑁)) → 𝑚 ≤ ((𝑇 + 1) · 𝑁))
156	155	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 𝑚 ≤ ((𝑇 + 1) · 𝑁))
157	136	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑁 ∈ ℕ0)
158	16	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑆 ∈ ℕ0)
159	129	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑍 ∈ ℂ)
160	20	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
161	23	peano2zd 11485	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇 + 1) ∈ ℤ)
162		eluz 11701	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 + 1) ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑆 ∈ (ℤ≥‘(𝑇 + 1)) ↔ (𝑇 + 1) ≤ 𝑆))
163	162	biimpar 502	. . . . . . . . . . . . . . 15 ⊢ ((((𝑇 + 1) ∈ ℤ ∧ 𝑆 ∈ ℤ) ∧ (𝑇 + 1) ≤ 𝑆) → 𝑆 ∈ (ℤ≥‘(𝑇 + 1)))
164	161, 24, 30, 163	syl21anc 1325	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘(𝑇 + 1)))
165		fzoss2 12496	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (ℤ≥‘(𝑇 + 1)) → (0..^(𝑇 + 1)) ⊆ (0..^𝑆))
166	164, 165	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^(𝑇 + 1)) ⊆ (0..^𝑆))
167	166	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → (0..^(𝑇 + 1)) ⊆ (0..^𝑆))
168		simplr 792	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ (0..^(𝑇 + 1)))
169	167, 168	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ (0..^𝑆))
170		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
171	157, 158, 159, 160, 169, 170	breprexplemb 30709	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑥 ∈ (0..^(𝑇 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
172	136, 123, 131, 156, 171	breprexplema 30708	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)))
173	172	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = (Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)))
174	128	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
175	152, 174	mulcld 10060	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) ∈ ℂ)
176	127, 175	fsumcl 14464	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) ∈ ℂ)
177	125, 132, 176	fsummulc1 14517	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)))
178	127, 133, 175	fsummulc1 14517	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)))
179	133	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → (𝑍↑𝑚) ∈ ℂ)
180	152, 174, 179	mulassd 10063	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))) → ((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
181	180	sumeq2dv 14433	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
182	178, 181	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
183	182	sumeq2dv 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑚)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
184	173, 177, 183	3eqtrd 2660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
185	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (1...𝑁) ⊆ ℕ)
186		1nn0 11308	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
187	186	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → 1 ∈ ℕ0)
188	123, 187	nn0addcld 11355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (𝑇 + 1) ∈ ℕ0)
189	185, 117, 188, 125	reprfi 30694	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → ((1...𝑁)(repr‘(𝑇 + 1))𝑚) ∈ Fin)
190		fzofi 12773	. . . . . . . . . 10 ⊢ (0..^(𝑇 + 1)) ∈ Fin
191	190	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) → (0..^(𝑇 + 1)) ∈ Fin)
192	136	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑁 ∈ ℕ0)
193	16	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑆 ∈ ℕ0)
194	129	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑍 ∈ ℂ)
195	20	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
196	166	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) → (0..^(𝑇 + 1)) ⊆ (0..^𝑆))
197	196	sselda 3603	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑎 ∈ (0..^𝑆))
198	39	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → (1...𝑁) ⊆ ℕ)
199	117	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑚 ∈ ℤ)
200	188	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → (𝑇 + 1) ∈ ℕ0)
201		simplr 792	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚))
202	198, 199, 200, 201	reprf 30690	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑑:(0..^(𝑇 + 1))⟶(1...𝑁))
203		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → 𝑎 ∈ (0..^(𝑇 + 1)))
204	202, 203	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → (𝑑‘𝑎) ∈ (1...𝑁))
205	39, 204	sseldi 3601	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → (𝑑‘𝑎) ∈ ℕ)
206	192, 193, 194, 195, 197, 205	breprexplemb 30709	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) ∧ 𝑎 ∈ (0..^(𝑇 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
207	191, 206	fprodcl 14682	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)) → ∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
208	189, 132, 207	fsummulc1 14517	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → (Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))
209	154, 184, 208	3eqtr2rd 2663	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...((𝑇 + 1) · 𝑁))) → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
210	209	sumeq2dv 14433	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
211		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑛 − 𝑏) = (𝑚 − 𝑏))
212	211	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏)) = ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏)))
213	212	sumeq1d 14431	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)))
214		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑍↑𝑛) = (𝑍↑𝑚))
215	214	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛)) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)))
216	213, 215	oveq12d 6668	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
217	216	adantr 481	. . . . . . 7 ⊢ ((𝑛 = 𝑚 ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
218	217	sumeq2dv 14433	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚))))
219	218	cbvsumv 14426	. . . . 5 ⊢ Σ𝑛 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑚 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)))
220	210, 219	syl6eqr 2674	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) = Σ𝑛 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))))
221	1, 14	nn0mulcld 11356	. . . . . 6 ⊢ (𝜑 → (𝑇 · 𝑁) ∈ ℕ0)
222		oveq2 6658	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑏) → ((1...𝑁)(repr‘𝑇)𝑚) = ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏)))
223	222	sumeq1d 14431	. . . . . . 7 ⊢ (𝑚 = (𝑛 − 𝑏) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)))
224		oveq1 6657	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 𝑏) → (𝑚 + 𝑏) = ((𝑛 − 𝑏) + 𝑏))
225	224	oveq2d 6666	. . . . . . . 8 ⊢ (𝑚 = (𝑛 − 𝑏) → (𝑍↑(𝑚 + 𝑏)) = (𝑍↑((𝑛 − 𝑏) + 𝑏)))
226	225	oveq2d 6666	. . . . . . 7 ⊢ (𝑚 = (𝑛 − 𝑏) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏))))
227	223, 226	oveq12d 6668	. . . . . 6 ⊢ (𝑚 = (𝑛 − 𝑏) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
228	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
229		uzssz 11707	. . . . . . . . . 10 ⊢ (ℤ≥‘-𝑏) ⊆ ℤ
230		simp2 1062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑚 ∈ (ℤ≥‘-𝑏))
231	229, 230	sseldi 3601	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑚 ∈ ℤ)
232	1	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑇 ∈ ℕ0)
233	57	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
234	228, 231, 232, 233	reprfi 30694	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑇)𝑚) ∈ Fin)
235	9	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (0..^𝑇) ∈ Fin)
236	58	3adant2 1080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑁 ∈ ℕ0)
237	236	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑁 ∈ ℕ0)
238	59	3adant2 1080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
239	238	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑆 ∈ ℕ0)
240	60	3adant2 1080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
241	240	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑍 ∈ ℂ)
242	61	3adant2 1080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
243	242	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
244	36	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (0..^𝑇) ⊆ (0..^𝑆))
245	244	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (0..^𝑇) ⊆ (0..^𝑆))
246	245	sselda 3603	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑆))
247	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (1...𝑁) ⊆ ℕ)
248	231	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑚 ∈ ℤ)
249	232	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑇 ∈ ℕ0)
250		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚))
251	247, 248, 249, 250	reprf 30690	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑑:(0..^𝑇)⟶(1...𝑁))
252	251	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑑:(0..^𝑇)⟶(1...𝑁))
253		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → 𝑎 ∈ (0..^𝑇))
254	252, 253	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ (1...𝑁))
255	39, 254	sseldi 3601	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → (𝑑‘𝑎) ∈ ℕ)
256	237, 239, 241, 243, 246, 255	breprexplemb 30709	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) ∧ 𝑎 ∈ (0..^𝑇)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
257	235, 256	fprodcl 14682	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
258	234, 257	fsumcl 14464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
259	70	3adant2 1080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑇 ∈ (0..^𝑆))
260	72	3adant2 1080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ)
261	236, 238, 240, 242, 259, 260	breprexplemb 30709	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
262	231	zcnd 11483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑚 ∈ ℂ)
263		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
264	119, 263	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
265	264	zcnd 11483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℂ)
266	262, 265	subnegd 10399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (𝑚 − -𝑏) = (𝑚 + 𝑏))
267	264	znegcld 11484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → -𝑏 ∈ ℤ)
268		eluzle 11700	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ≥‘-𝑏) → -𝑏 ≤ 𝑚)
269	230, 268	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → -𝑏 ≤ 𝑚)
270		znn0sub 11424	. . . . . . . . . . . 12 ⊢ ((-𝑏 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (-𝑏 ≤ 𝑚 ↔ (𝑚 − -𝑏) ∈ ℕ0))
271	270	biimpa 501	. . . . . . . . . . 11 ⊢ (((-𝑏 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ -𝑏 ≤ 𝑚) → (𝑚 − -𝑏) ∈ ℕ0)
272	267, 231, 269, 271	syl21anc 1325	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (𝑚 − -𝑏) ∈ ℕ0)
273	266, 272	eqeltrrd 2702	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (𝑚 + 𝑏) ∈ ℕ0)
274	240, 273	expcld 13008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑(𝑚 + 𝑏)) ∈ ℂ)
275	261, 274	mulcld 10060	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))) ∈ ℂ)
276	258, 275	mulcld 10060	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘-𝑏) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) ∈ ℂ)
277	58	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑁 ∈ ℕ0)
278		ssid 3624	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (1...𝑁)
279	278	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (1...𝑁) ⊆ (1...𝑁))
280		fzssz 12343	. . . . . . . . . . . . 13 ⊢ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁)) ⊆ ℤ
281		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁)))
282	280, 281	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑛 ∈ ℤ)
283		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑏 ∈ (1...𝑁))
284	119, 283	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑏 ∈ ℤ)
285	282, 284	zsubcld 11487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (𝑛 − 𝑏) ∈ ℤ)
286	1	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑇 ∈ ℕ0)
287	25	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑇 ∈ ℝ)
288	277	nn0red 11352	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑁 ∈ ℝ)
289	287, 288	remulcld 10070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (𝑇 · 𝑁) ∈ ℝ)
290	284	zred 11482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑏 ∈ ℝ)
291	221	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑇 · 𝑁) ∈ ℕ0)
292	291, 74	nn0addcld 11355	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((𝑇 · 𝑁) + 𝑏) ∈ ℕ0)
293	186	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 1 ∈ ℕ0)
294	292, 293	nn0addcld 11355	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (((𝑇 · 𝑁) + 𝑏) + 1) ∈ ℕ0)
295		fz2ssnn0 29547	. . . . . . . . . . . . . . 15 ⊢ ((((𝑇 · 𝑁) + 𝑏) + 1) ∈ ℕ0 → ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁)) ⊆ ℕ0)
296	294, 295	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁)) ⊆ ℕ0)
297	296	sselda 3603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑛 ∈ ℕ0)
298	297	nn0red 11352	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑛 ∈ ℝ)
299	23	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑇 ∈ ℤ)
300	277	nn0zd 11480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑁 ∈ ℤ)
301	299, 300	zmulcld 11488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (𝑇 · 𝑁) ∈ ℤ)
302	301, 284	zaddcld 11486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((𝑇 · 𝑁) + 𝑏) ∈ ℤ)
303		elfzle1 12344	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁)) → (((𝑇 · 𝑁) + 𝑏) + 1) ≤ 𝑛)
304	281, 303	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (((𝑇 · 𝑁) + 𝑏) + 1) ≤ 𝑛)
305		zltp1le 11427	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 · 𝑁) + 𝑏) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑇 · 𝑁) + 𝑏) < 𝑛 ↔ (((𝑇 · 𝑁) + 𝑏) + 1) ≤ 𝑛))
306	305	biimpar 502	. . . . . . . . . . . . 13 ⊢ (((((𝑇 · 𝑁) + 𝑏) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (((𝑇 · 𝑁) + 𝑏) + 1) ≤ 𝑛) → ((𝑇 · 𝑁) + 𝑏) < 𝑛)
307	302, 282, 304, 306	syl21anc 1325	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((𝑇 · 𝑁) + 𝑏) < 𝑛)
308		ltaddsub 10502	. . . . . . . . . . . . 13 ⊢ (((𝑇 · 𝑁) ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (((𝑇 · 𝑁) + 𝑏) < 𝑛 ↔ (𝑇 · 𝑁) < (𝑛 − 𝑏)))
309	308	biimpa 501	. . . . . . . . . . . 12 ⊢ ((((𝑇 · 𝑁) ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ ((𝑇 · 𝑁) + 𝑏) < 𝑛) → (𝑇 · 𝑁) < (𝑛 − 𝑏))
310	289, 290, 298, 307, 309	syl31anc 1329	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (𝑇 · 𝑁) < (𝑛 − 𝑏))
311	277, 279, 285, 286, 310	reprgt 30699	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏)) = ∅)
312	311	sumeq1d 14431	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ∅ ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)))
313		sum0 14452	. . . . . . . . 9 ⊢ Σ𝑑 ∈ ∅ ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = 0
314	312, 313	syl6eq 2672	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = 0)
315	314	oveq1d 6665	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = (0 · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
316	73	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
317	60	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑍 ∈ ℂ)
318	282	zcnd 11483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑛 ∈ ℂ)
319	284	zcnd 11483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → 𝑏 ∈ ℂ)
320	318, 319	npcand 10396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((𝑛 − 𝑏) + 𝑏) = 𝑛)
321	320, 297	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → ((𝑛 − 𝑏) + 𝑏) ∈ ℕ0)
322	317, 321	expcld 13008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (𝑍↑((𝑛 − 𝑏) + 𝑏)) ∈ ℂ)
323	316, 322	mulcld 10060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏))) ∈ ℂ)
324	323	mul02d 10234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (0 · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = 0)
325	315, 324	eqtrd 2656	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ ((((𝑇 · 𝑁) + 𝑏) + 1)...((𝑇 · 𝑁) + 𝑁))) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = 0)
326	39	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (1...𝑁) ⊆ ℕ)
327		fzossfz 12488	. . . . . . . . . . . . . 14 ⊢ (0..^𝑏) ⊆ (0...𝑏)
328		fzssz 12343	. . . . . . . . . . . . . 14 ⊢ (0...𝑏) ⊆ ℤ
329	327, 328	sstri 3612	. . . . . . . . . . . . 13 ⊢ (0..^𝑏) ⊆ ℤ
330		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 ∈ (0..^𝑏))
331	329, 330	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 ∈ ℤ)
332		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑏 ∈ (1...𝑁))
333	119, 332	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑏 ∈ ℤ)
334	331, 333	zsubcld 11487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (𝑛 − 𝑏) ∈ ℤ)
335	1	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑇 ∈ ℕ0)
336	334	zred 11482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (𝑛 − 𝑏) ∈ ℝ)
337		0red 10041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 0 ∈ ℝ)
338	25	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑇 ∈ ℝ)
339		elfzolt2 12479	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (0..^𝑏) → 𝑛 < 𝑏)
340	339	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 < 𝑏)
341	331	zred 11482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 ∈ ℝ)
342	333	zred 11482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑏 ∈ ℝ)
343	341, 342	sublt0d 10653	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → ((𝑛 − 𝑏) < 0 ↔ 𝑛 < 𝑏))
344	340, 343	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (𝑛 − 𝑏) < 0)
345	62	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 0 ≤ 𝑇)
346	336, 337, 338, 344, 345	ltletrd 10197	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (𝑛 − 𝑏) < 𝑇)
347	326, 334, 335, 346	reprlt 30697	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏)) = ∅)
348	347	sumeq1d 14431	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ∅ ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)))
349	348, 313	syl6eq 2672	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) = 0)
350	349	oveq1d 6665	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = (0 · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
351	73	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
352	60	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑍 ∈ ℂ)
353	341	recnd 10068	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 ∈ ℂ)
354	342	recnd 10068	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑏 ∈ ℂ)
355	353, 354	npcand 10396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → ((𝑛 − 𝑏) + 𝑏) = 𝑛)
356		fzo0ssnn0 12548	. . . . . . . . . . . 12 ⊢ (0..^𝑏) ⊆ ℕ0
357	356, 330	sseldi 3601	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → 𝑛 ∈ ℕ0)
358	355, 357	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → ((𝑛 − 𝑏) + 𝑏) ∈ ℕ0)
359	352, 358	expcld 13008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (𝑍↑((𝑛 − 𝑏) + 𝑏)) ∈ ℂ)
360	351, 359	mulcld 10060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏))) ∈ ℂ)
361	360	mul02d 10234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (0 · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = 0)
362	350, 361	eqtrd 2656	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑛 ∈ (0..^𝑏)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))) = 0)
363	221, 14, 227, 276, 325, 362	fsum2dsub 30685	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = Σ𝑛 ∈ (0...((𝑇 · 𝑁) + 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
364		nn0sscn 11297	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℂ
365	364, 1	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ)
366	364, 14	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
367	365, 366	adddirp1d 10066	. . . . . . 7 ⊢ (𝜑 → ((𝑇 + 1) · 𝑁) = ((𝑇 · 𝑁) + 𝑁))
368	367	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (0...((𝑇 + 1) · 𝑁)) = (0...((𝑇 · 𝑁) + 𝑁)))
369	130, 364	sstri 3612	. . . . . . . . . . . . 13 ⊢ (0...((𝑇 + 1) · 𝑁)) ⊆ ℂ
370		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑛 ∈ (0...((𝑇 + 1) · 𝑁)))
371	369, 370	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑛 ∈ ℂ)
372	44, 364	sstri 3612	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℂ
373		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
374	372, 373	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℂ)
375	371, 374	npcand 10396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → ((𝑛 − 𝑏) + 𝑏) = 𝑛)
376	375	eqcomd 2628	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑛 = ((𝑛 − 𝑏) + 𝑏))
377	376	oveq2d 6666	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑛) = (𝑍↑((𝑛 − 𝑏) + 𝑏)))
378	377	oveq2d 6666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛)) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏))))
379	378	oveq2d 6666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
380	379	sumeq2dv 14433	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...((𝑇 + 1) · 𝑁))) → Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
381	368, 380	sumeq12dv 14437	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))) = Σ𝑛 ∈ (0...((𝑇 · 𝑁) + 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑((𝑛 − 𝑏) + 𝑏)))))
382	363, 381	eqtr4d 2659	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = Σ𝑛 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)(𝑛 − 𝑏))∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑛))))
383	107	adantlr 751	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
384	110	adantlr 751	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (𝑍↑𝑚) ∈ ℂ)
385	76	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
386	385	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)) ∈ ℂ)
387	383, 384, 386	mulassd 10063	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝑍↑𝑚) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)))))
388	73	ad4ant13 1292	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
389	75	ad4ant13 1292	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (𝑍↑𝑏) ∈ ℂ)
390	384, 388, 389	mulassd 10063	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (((𝑍↑𝑚) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑏)) = ((𝑍↑𝑚) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
391	388, 384, 389	mulassd 10063	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)) · (𝑍↑𝑏)) = (((𝐿‘𝑇)‘𝑏) · ((𝑍↑𝑚) · (𝑍↑𝑏))))
392	384, 388	mulcomd 10061	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((𝑍↑𝑚) · ((𝐿‘𝑇)‘𝑏)) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)))
393	392	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (((𝑍↑𝑚) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑏)) = ((((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑚)) · (𝑍↑𝑏)))
394	108	adantlr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑍 ∈ ℂ)
395	74	ad4ant13 1292	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑏 ∈ ℕ0)
396	109	adantlr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → 𝑚 ∈ ℕ0)
397	394, 395, 396	expaddd 13010	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (𝑍↑(𝑚 + 𝑏)) = ((𝑍↑𝑚) · (𝑍↑𝑏)))
398	397	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))) = (((𝐿‘𝑇)‘𝑏) · ((𝑍↑𝑚) · (𝑍↑𝑏))))
399	391, 393, 398	3eqtr4d 2666	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (((𝑍↑𝑚) · ((𝐿‘𝑇)‘𝑏)) · (𝑍↑𝑏)) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))))
400	390, 399	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((𝑍↑𝑚) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))))
401	400	oveq2d 6666	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝑍↑𝑚) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏)))) = (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))))
402	387, 401	eqtrd 2656	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → ((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))))
403	402	sumeq2dv 14433	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))))
404	89	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑇)𝑚) ∈ Fin)
405	111	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)) → (∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) ∈ ℂ)
406	404, 385, 405	fsummulc1 14517	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)((∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
407	73	adantlr 751	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑇)‘𝑏) ∈ ℂ)
408	60	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
409	85	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑚 ∈ ℕ0)
410	74	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
411	409, 410	nn0addcld 11355	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (𝑚 + 𝑏) ∈ ℕ0)
412	408, 411	expcld 13008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑(𝑚 + 𝑏)) ∈ ℂ)
413	407, 412	mulcld 10060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏))) ∈ ℂ)
414	404, 413, 383	fsummulc1 14517	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))))
415	403, 406, 414	3eqtr4rd 2667	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) ∧ 𝑏 ∈ (1...𝑁)) → (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = (Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
416	415	sumeq2dv 14433	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑇 · 𝑁))) → Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
417	416	sumeq2dv 14433	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑(𝑚 + 𝑏)))) = Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))))
418	220, 382, 417	3eqtr2rd 2663	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑏 ∈ (1...𝑁)(Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)) · (((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))
419	80, 113, 418	3eqtr2d 2662	. 2 ⊢ (𝜑 → (∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) · Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑇)‘𝑏) · (𝑍↑𝑏))) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))
420	6, 78, 419	3eqtrd 2660	1 ⊢ (𝜑 → ∏𝑎 ∈ (0..^(𝑇 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   − cmin 10266  -cneg 10267  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  ↑cexp 12860  Σcsu 14416  ∏cprod 14635  reprcrepr 30686

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-disj 4621  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  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-ico 12181  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  df-prod 14636  df-repr 30687

This theorem is referenced by:  breprexp  30711