Theorem breprexpnat 31900

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 6678	. . . . . 6 ⊢ ((𝟭‘ℕ)‘𝐴) ∈ V
5	4	fconst 6560	. . . . 5 ⊢ ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6		nnex 11638	. . . . . . . . 9 ⊢ ℕ ∈ V
7		breprexpnat.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
8		indf 31269	. . . . . . . . 9 ⊢ ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
9	6, 7, 8	sylancr 589	. . . . . . . 8 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10		0cn 10627	. . . . . . . . 9 ⊢ 0 ∈ ℂ
11		ax-1cn 10589	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		prssi 4748	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
13	10, 11, 12	mp2an 690	. . . . . . . 8 ⊢ {0, 1} ⊆ ℂ
14		fss 6522	. . . . . . . 8 ⊢ ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
15	9, 13, 14	sylancl 588	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16		cnex 10612	. . . . . . . 8 ⊢ ℂ ∈ V
17	16, 6	elmap 8429	. . . . . . 7 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
18	15, 17	sylibr 236	. . . . . 6 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
19	4	snss 4712	. . . . . 6 ⊢ (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
20	18, 19	sylib 220	. . . . 5 ⊢ (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21		fss 6522	. . . . 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 589	. . . 4 ⊢ (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
23	1, 2, 3, 22	breprexp 31899	. . 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 6961	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
25	24	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
26	25	fveq1d 6667	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
27	26	oveq1d 7165	. . . . . . 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 15054	. . . . . 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 13334	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32		fz1ssnn 12932	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
33	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
34	7	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
35	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36		nnssnn0 11894	. . . . . . . . . 10 ⊢ ℕ ⊆ ℕ0
37	32, 36	sstri 3976	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ0
38		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
39	37, 38	sseldi 3965	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
40	35, 39	expcld 13504	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
41	29, 31, 33, 34, 40	indsumin 31276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42		incom 4178	. . . . . . . 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 15052	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
45	28, 41, 44	3eqtrd 2860	. . . . 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 15271	. . . 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 13336	. . . . . 6 ⊢ (0..^𝑆) ∈ Fin
48	47	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ Fin)
49		inss2 4206	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50		ssfi 8732	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
51	30, 49, 50	mp2an 690	. . . . . . 7 ⊢ (𝐴 ∩ (1...𝑁)) ∈ Fin
52	51	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
53	3	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
54	49, 37	sstri 3976	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
56	54, 55	sseldi 3965	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
57	53, 56	expcld 13504	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍↑𝑏) ∈ ℂ)
58	52, 57	fsumcl 15084	. . . . 5 ⊢ (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ)
59		fprodconst 15326	. . . . 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 586	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
61		hashfzo0 13785	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
62	2, 61	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
63	62	oveq2d 7166	. . . 4 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
64	46, 60, 63	3eqtrd 2860	. . 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 12903	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℤ
67		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
68	66, 67	sseldi 3965	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
69	2	adantr 483	. . . . . . 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 31882	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
72	3	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73		fz0ssnn0 12996	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℕ0
74	73, 67	sseldi 3965	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
75	72, 74	expcld 13504	. . . . . 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 728	. . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
80	69	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
82	78, 79, 80, 81	reprf 31878	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
83	82	ffvelrnda 6846	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
84	32, 83	sseldi 3965	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
85	77, 84	ffvelrnd 6847	. . . . . . . 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	sseldi 3965	. . . . . . 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 15300	. . . . . 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 15134	. . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
90	89, 68, 69, 70, 65	hashreprin 31886	. . . . . 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 7165	. . . . 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 6667	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐‘𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
93	92	adantl 484	. . . . . . . . 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 15271	. . . . . . . 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 483	. . . . . . 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 7165	. . . . . 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 15054	. . . . 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 2867	. . . 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 15054	. . 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 2864	. 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 7160	. 2 ⊢ (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103		breprexpnat.r	. . . . 5 ⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104	103	oveq1i 7160	. . . 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 15050	. 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 2881	1 ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3495   ∩ cin 3935   ⊆ wss 3936  {csn 4561  {cpr 4563   × cxp 5548  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150   ↑_m cmap 8400  Fincfn 8503  ℂcc 10529  0cc0 10531  1c1 10532   · cmul 10536  ℕcn 11632  ℕ₀cn0 11891  ℤcz 11975  ...cfz 12886  ..^cfzo 13027  ↑cexp 13423  ♯chash 13684  Σcsu 15036  ∏cprod 15253  𝟭cind 31264  reprcrepr 31874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-disj 5025  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-ico 12738  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-sum 15037  df-prod 15254  df-ind 31265  df-repr 31875

This theorem is referenced by: (None)