Theorem breprexpnat 34772

Description: Express the 𝑆 th power of the finite series in terms of the number of representations of integers 𝑚 as sums of 𝑆 terms of elements of 𝐴, bounded by 𝑁. Proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
breprexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
breprexp.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
breprexp.z	⊢ (𝜑 → 𝑍 ∈ ℂ)
breprexpnat.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
breprexpnat.p	⊢ 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)
breprexpnat.r	⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))

Assertion

Ref	Expression
breprexpnat	⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Distinct variable groups:   𝑚,𝑁   𝑆,𝑚   𝑚,𝑍   𝐴,𝑏,𝑚   𝑁,𝑏   𝑆,𝑏   𝑍,𝑏   𝜑,𝑏,𝑚

Allowed substitution hints:   𝑃(𝑚,𝑏)   𝑅(𝑚,𝑏)

Proof of Theorem breprexpnat

Dummy variables 𝑐 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breprexp.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		breprexp.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
3		breprexp.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ ℂ)
4		fvex 6848	. . . . . 6 ⊢ ((𝟭‘ℕ)‘𝐴) ∈ V
5	4	fconst 6721	. . . . 5 ⊢ ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6		nnex 12155	. . . . . . . . 9 ⊢ ℕ ∈ V
7		breprexpnat.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
8		indf 32915	. . . . . . . . 9 ⊢ ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
9	6, 7, 8	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10		0cn 11128	. . . . . . . . 9 ⊢ 0 ∈ ℂ
11		ax-1cn 11088	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		prssi 4778	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
13	10, 11, 12	mp2an 693	. . . . . . . 8 ⊢ {0, 1} ⊆ ℂ
14		fss 6679	. . . . . . . 8 ⊢ ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
15	9, 13, 14	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16		cnex 11111	. . . . . . . 8 ⊢ ℂ ∈ V
17	16, 6	elmap 8813	. . . . . . 7 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
18	15, 17	sylibr 234	. . . . . 6 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
19	4	snss 4742	. . . . . 6 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
20	18, 19	sylib 218	. . . . 5 ⊢ (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21		fss 6679	. . . . 5 ⊢ ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)} ∧ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ)) → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
22	5, 20, 21	sylancr 588	. . . 4 ⊢ (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
23	1, 2, 3, 22	breprexp 34771	. . 3 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
24	4	fvconst2 7152	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
25	24	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
26	25	fveq1d 6837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
27	26	oveq1d 7375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)))
28	27	sumeq2dv 15629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)))
29	6	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → ℕ ∈ V)
30		fzfi 13899	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32		fz1ssnn 13475	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
33	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
34	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
35	3	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36		nnssnn0 12408	. . . . . . . . . 10 ⊢ ℕ ⊆ ℕ0
37	32, 36	sstri 3944	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ0
38		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
39	37, 38	sselid 3932	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
40	35, 39	expcld 14073	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
41	29, 31, 33, 34, 40	indsumin 32924	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42		incom 4162	. . . . . . . 8 ⊢ ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
43	42	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
44	43	sumeq1d 15627	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
45	28, 41, 44	3eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
46	45	prodeq2dv 15849	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
47		fzofi 13901	. . . . . 6 ⊢ (0..^𝑆) ∈ Fin
48	47	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ Fin)
49		inss2 4191	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50		ssfi 9101	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
51	30, 49, 50	mp2an 693	. . . . . . 7 ⊢ (𝐴 ∩ (1...𝑁)) ∈ Fin
52	51	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
53	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
54	49, 37	sstri 3944	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
56	54, 55	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
57	53, 56	expcld 14073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍↑𝑏) ∈ ℂ)
58	52, 57	fsumcl 15660	. . . . 5 ⊢ (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ)
59		fprodconst 15905	. . . . 5 ⊢ (((0..^𝑆) ∈ Fin ∧ Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ) → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
60	48, 58, 59	syl2anc 585	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
61		hashfzo0 14357	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
62	2, 61	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
63	62	oveq2d 7376	. . . 4 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
64	46, 60, 63	3eqtrd 2776	. . 3 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
65	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
66		fzssz 13446	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℤ
67		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
68	66, 67	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
69	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
70	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
71	65, 68, 69, 70	reprfi 34754	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
72	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73		fz0ssnn0 13542	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℕ0
74	73, 67	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
75	72, 74	expcld 14073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑍↑𝑚) ∈ ℂ)
76	47	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
77	9	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
78	32	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
79	68	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
80	69	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
82	78, 79, 80, 81	reprf 34750	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
83	82	ffvelcdmda 7031	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
84	32, 83	sselid 3932	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
85	77, 84	ffvelcdmd 7032	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) ∈ {0, 1})
86	13, 85	sselid 3932	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) ∈ ℂ)
87	76, 86	fprodcl 15879	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) ∈ ℂ)
88	71, 75, 87	fsummulc1 15712	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
89	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
90	89, 68, 69, 70, 65	hashreprin 34758	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
91	90	oveq1d 7375	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
92	24	fveq1d 6837	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
93	92	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑎 ∈ (0..^𝑆)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
94	93	prodeq2dv 15849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
95	94	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
96	95	oveq1d 7375	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
97	96	sumeq2dv 15629	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
98	88, 91, 97	3eqtr4rd 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚)))
99	98	sumeq2dv 15629	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚)))
100	23, 64, 99	3eqtr3d 2780	. 2 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚)))
101		breprexpnat.p	. . 3 ⊢ 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)
102	101	oveq1i 7370	. 2 ⊢ (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103		breprexpnat.r	. . . . 5 ⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104	103	oveq1i 7370	. . . 4 ⊢ (𝑅 · (𝑍↑𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚))
105	104	a1i 11	. . 3 ⊢ (𝑚 ∈ (0...(𝑆 · 𝑁)) → (𝑅 · (𝑍↑𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚)))
106	105	sumeq2i 15625	. 2 ⊢ Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍↑𝑚))
107	100, 102, 106	3eqtr4g 2797	1 ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  {csn 4581  {cpr 4583   × cxp 5623  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767  Fincfn 8887  ℂcc 11028  0cc0 11030  1c1 11031   · cmul 11035  ℕcn 12149  ℕ₀cn0 12405  ℤcz 12492  ...cfz 13427  ..^cfzo 13574  ↑cexp 13988  ♯chash 14257  Σcsu 15613  ∏cprod 15830  𝟭cind 32910  reprcrepr 34746

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-rp 12910  df-ico 13271  df-fz 13428  df-fzo 13575  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-sum 15614  df-prod 15831  df-ind 32911  df-repr 34747

This theorem is referenced by: (None)