Theorem breprexpnat 34654

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 6841	. . . . . 6 ⊢ ((𝟭‘ℕ)‘𝐴) ∈ V
5	4	fconst 6715	. . . . 5 ⊢ ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6		nnex 12137	. . . . . . . . 9 ⊢ ℕ ∈ V
7		breprexpnat.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
8		indf 32843	. . . . . . . . 9 ⊢ ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
9	6, 7, 8	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10		0cn 11110	. . . . . . . . 9 ⊢ 0 ∈ ℂ
11		ax-1cn 11070	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		prssi 4772	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
13	10, 11, 12	mp2an 692	. . . . . . . 8 ⊢ {0, 1} ⊆ ℂ
14		fss 6673	. . . . . . . 8 ⊢ ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
15	9, 13, 14	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16		cnex 11093	. . . . . . . 8 ⊢ ℂ ∈ V
17	16, 6	elmap 8801	. . . . . . 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 4736	. . . . . 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 6673	. . . . 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 587	. . . 4 ⊢ (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
23	1, 2, 3, 22	breprexp 34653	. . 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 7144	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
25	24	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
26	25	fveq1d 6830	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
27	26	oveq1d 7367	. . . . . . 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 15615	. . . . . 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 13885	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32		fz1ssnn 13461	. . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36		nnssnn0 12390	. . . . . . . . . 10 ⊢ ℕ ⊆ ℕ0
37	32, 36	sstri 3939	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ0
38		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
39	37, 38	sselid 3927	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
40	35, 39	expcld 14059	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
41	29, 31, 33, 34, 40	indsumin 32850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42		incom 4158	. . . . . . . 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 15613	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
45	28, 41, 44	3eqtrd 2770	. . . . 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 15835	. . . 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 13887	. . . . . 6 ⊢ (0..^𝑆) ∈ Fin
48	47	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ Fin)
49		inss2 4187	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50		ssfi 9088	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
51	30, 49, 50	mp2an 692	. . . . . . 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 3939	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
56	54, 55	sselid 3927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
57	53, 56	expcld 14059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍↑𝑏) ∈ ℂ)
58	52, 57	fsumcl 15646	. . . . 5 ⊢ (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ)
59		fprodconst 15891	. . . . 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 584	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
61		hashfzo0 14343	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
62	2, 61	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
63	62	oveq2d 7368	. . . 4 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
64	46, 60, 63	3eqtrd 2770	. . 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 13432	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℤ
67		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
68	66, 67	sselid 3927	. . . . . . 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 34636	. . . . . 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 13528	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℕ0
74	73, 67	sselid 3927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
75	72, 74	expcld 14059	. . . . . 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 730	. . . . . . . . 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 34632	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
83	82	ffvelcdmda 7023	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
84	32, 83	sselid 3927	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
85	77, 84	ffvelcdmd 7024	. . . . . . . 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 3927	. . . . . . 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 15865	. . . . . 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 15698	. . . . 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 34640	. . . . . 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 7367	. . . . 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 6830	. . . . . . . . . 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 15835	. . . . . . . 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 7367	. . . . . 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 15615	. . . . 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 2777	. . . 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 15615	. . 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 2774	. 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 7362	. 2 ⊢ (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103		breprexpnat.r	. . . . 5 ⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104	103	oveq1i 7362	. . . 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 15611	. 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 2791	1 ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  {csn 4575  {cpr 4577   × cxp 5617  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352   ↑_m cmap 8756  Fincfn 8875  ℂcc 11010  0cc0 11012  1c1 11013   · cmul 11017  ℕcn 12131  ℕ₀cn0 12387  ℤcz 12474  ...cfz 13413  ..^cfzo 13560  ↑cexp 13974  ♯chash 14243  Σcsu 15599  ∏cprod 15816  𝟭cind 32838  reprcrepr 34628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9537  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089  ax-pre-sup 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-disj 5061  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9332  df-oi 9402  df-card 9838  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-div 11781  df-nn 12132  df-2 12194  df-3 12195  df-n0 12388  df-z 12475  df-uz 12739  df-rp 12897  df-ico 13257  df-fz 13414  df-fzo 13561  df-seq 13915  df-exp 13975  df-hash 14244  df-cj 15012  df-re 15013  df-im 15014  df-sqrt 15148  df-abs 15149  df-clim 15401  df-sum 15600  df-prod 15817  df-ind 32839  df-repr 34629

This theorem is referenced by: (None)