Theorem breprexpnat 34875

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 6865	. . . . . 6 ⊢ ((𝟭‘ℕ)‘𝐴) ∈ V
5	4	fconst 6735	. . . . 5 ⊢ ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6		nnex 12202	. . . . . . . . 9 ⊢ ℕ ∈ V
7		breprexpnat.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
8		indf 12187	. . . . . . . . 9 ⊢ ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
9	6, 7, 8	sylancr 595	. . . . . . . 8 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10		0cn 11157	. . . . . . . . 9 ⊢ 0 ∈ ℂ
11		ax-1cn 11117	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		prssi 4769	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
13	10, 11, 12	mp2an 700	. . . . . . . 8 ⊢ {0, 1} ⊆ ℂ
14		fss 6693	. . . . . . . 8 ⊢ ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
15	9, 13, 14	sylancl 594	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16		cnex 11140	. . . . . . . 8 ⊢ ℂ ∈ V
17	16, 6	elmap 8838	. . . . . . 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 4733	. . . . . 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 6693	. . . . 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 595	. . . 4 ⊢ (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
23	1, 2, 3, 22	breprexp 34874	. . 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 7173	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
25	24	ad2antlr 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
26	25	fveq1d 6854	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
27	26	oveq1d 7396	. . . . . . 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 15701	. . . . . 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 13971	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32		fz1ssnn 13546	. . . . . . . 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 734	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36		nnssnn0 12470	. . . . . . . . . 10 ⊢ ℕ ⊆ ℕ0
37	32, 36	sstri 3936	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ0
38		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
39	37, 38	sselid 3925	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
40	35, 39	expcld 14145	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍↑𝑏) ∈ ℂ)
41	29, 31, 33, 34, 40	indsumin 32989	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42		incom 4152	. . . . . . . 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 15699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
45	28, 41, 44	3eqtrd 2791	. . . . 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 15924	. . . 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 13973	. . . . . 6 ⊢ (0..^𝑆) ∈ Fin
48	47	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ Fin)
49		inss2 4180	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50		ssfi 9126	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
51	30, 49, 50	mp2an 700	. . . . . . 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 3936	. . . . . . . 8 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
56	54, 55	sselid 3925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
57	53, 56	expcld 14145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍↑𝑏) ∈ ℂ)
58	52, 57	fsumcl 15732	. . . . 5 ⊢ (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ ℂ)
59		fprodconst 15980	. . . . 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 592	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))))
61		hashfzo0 14429	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
62	2, 61	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
63	62	oveq2d 7397	. . . 4 ⊢ (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
64	46, 60, 63	3eqtrd 2791	. . 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 13517	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℤ
67		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
68	66, 67	sselid 3925	. . . . . . 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 34857	. . . . . 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 13613	. . . . . . . 8 ⊢ (0...(𝑆 · 𝑁)) ⊆ ℕ0
74	73, 67	sselid 3925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
75	72, 74	expcld 14145	. . . . . 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 738	. . . . . . . . 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 34853	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
83	82	ffvelcdmda 7050	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
84	32, 83	sselid 3925	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
85	77, 84	ffvelcdmd 7051	. . . . . . . 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 3925	. . . . . . 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 15954	. . . . . 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 15784	. . . . 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 34861	. . . . . 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 7396	. . . . 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 6854	. . . . . . . . . 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 15924	. . . . . . . 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 7396	. . . . . 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 15701	. . . . 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 2798	. . . 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 15701	. . 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 2795	. 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 7391	. 2 ⊢ (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103		breprexpnat.r	. . . . 5 ⊢ 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104	103	oveq1i 7391	. . . 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 15697	. 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 2812	1 ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  {csn 4572  {cpr 4574   × cxp 5634  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ↑_m cmap 8792  Fincfn 8912  ℂcc 11057  0cc0 11059  1c1 11060   · cmul 11064  𝟭cind 12181  ℕcn 12196  ℕ₀cn0 12467  ℤcz 12554  ...cfz 13498  ..^cfzo 13645  ↑cexp 14060  ♯chash 14329  Σcsu 15685  ∏cprod 15905  reprcrepr 34849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-disj 5058  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-pm 8795  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-sup 9374  df-oi 9444  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-ind 12182  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-rp 12980  df-ico 13341  df-fz 13499  df-fzo 13646  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-sum 15686  df-prod 15906  df-repr 34850

This theorem is referenced by: (None)