Theorem breprexpnat 34987

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 6894	. . . . . 6 ⊢ ((𝟭‘ℕ)‘𝐴) ∈ V
5	4	fconst 6764	. . . . 5 ⊢ ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6		nnex 12238	. . . . . . . . 9 ⊢ ℕ ∈ V
7		breprexpnat.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
8		indf 12223	. . . . . . . . 9 ⊢ ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
9	6, 7, 8	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10		0cn 11197	. . . . . . . . 9 ⊢ 0 ∈ ℂ
11		ax-1cn 11157	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		prssi 4786	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
13	10, 11, 12	mp2an 704	. . . . . . . 8 ⊢ {0, 1} ⊆ ℂ
14		fss 6722	. . . . . . . 8 ⊢ ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
15	9, 13, 14	sylancl 597	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16		cnex 11180	. . . . . . . 8 ⊢ ℂ ∈ V
17	16, 6	elmap 8868	. . . . . . 7 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
18	15, 17	sylibr 237	. . . . . 6 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
19	4	snss 4749	. . . . . 6 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
20	18, 19	sylib 221	. . . . 5 ⊢ (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21		fss 6722	. . . . 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 598	. . . 4 ⊢ (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
23	1, 2, 3, 22	breprexp 34986	. . 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 7202	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
25	24	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
26	25	fveq1d 6883	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
27	26	oveq1d 7425	. . . . . . 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 15752	. . . . . 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 14007	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32		fz1ssnn 13582	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
33	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
34	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
35	3	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36		nnssnn0 12506	. . . . . . . . . 10 ⊢ ℕ ⊆ ℕ0
37	32, 36	sstri 3945	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ0
38		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
39	37, 38	sselid 3934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
40	35, 39	expcld 14181	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
41	29, 31, 33, 34, 40	indsumin 33147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42		incom 4161	. . . . . . . 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 15750	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
45	28, 41, 44	3eqtrd 2800	. . . . 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 15975	. . . 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 14009	. . . . . 6 ⊢ (0..^𝑆) ∈ Fin
48	47	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ Fin)
49		inss2 4189	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50		ssfi 9156	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
51	30, 49, 50	mp2an 704	. . . . . . 7 ⊢ (𝐴 ∩ (1...𝑁)) ∈ Fin
52	51	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
53	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
54	49, 37	sstri 3945	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
56	54, 55	sselid 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
57	53, 56	expcld 14181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍↑𝑏) ∈ ℂ)
58	52, 57	fsumcl 15783	. . . . 5 ⊢ (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ)
59		fprodconst 16031	. . . . 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 595	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
61		hashfzo0 14466	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
62	2, 61	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
63	62	oveq2d 7426	. . . 4 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
64	46, 60, 63	3eqtrd 2800	. . 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 13553	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℤ
67		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
68	66, 67	sselid 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
69	2	adantr 485	. . . . . . 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 34969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
72	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73		fz0ssnn0 13649	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℕ0
74	73, 67	sselid 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
75	72, 74	expcld 14181	. . . . . 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 742	. . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
80	69	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
82	78, 79, 80, 81	reprf 34965	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
83	82	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
84	32, 83	sselid 3934	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
85	77, 84	ffvelcdmd 7080	. . . . . . . 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 3934	. . . . . . 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 16005	. . . . . 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 15835	. . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
90	89, 68, 69, 70, 65	hashreprin 34973	. . . . . 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 7425	. . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
93	92	adantl 486	. . . . . . . . 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 15975	. . . . . . . 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 485	. . . . . . 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 7425	. . . . . 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 15752	. . . . 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 2807	. . . 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 15752	. . 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 2804	. 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 7420	. 2 ⊢ (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103		breprexpnat.r	. . . . 5 ⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104	103	oveq1i 7420	. . . 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 15748	. 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 2821	1 ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  {csn 4588  {cpr 4590   × cxp 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8823  Fincfn 8942  ℂcc 11097  0cc0 11099  1c1 11100   · cmul 11104  𝟭cind 12217  ℕcn 12232  ℕ₀cn0 12503  ℤcz 12590  ...cfz 13534  ..^cfzo 13681  ↑cexp 14096  ♯chash 14365  Σcsu 15736  ∏cprod 15956  reprcrepr 34961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-ind 12218  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-ico 13377  df-fz 13535  df-fzo 13682  df-seq 14037  df-exp 14097  df-hash 14366  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-clim 15538  df-sum 15737  df-prod 15957  df-repr 34962

This theorem is referenced by: (None)