Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  breprexpnat Structured version   Visualization version   GIF version

Theorem breprexpnat 34649
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 (𝜑𝑁 ∈ ℕ0)
breprexp.s (𝜑𝑆 ∈ ℕ0)
breprexp.z (𝜑𝑍 ∈ ℂ)
breprexpnat.a (𝜑𝐴 ⊆ ℕ)
breprexpnat.p 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)
breprexpnat.r 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
Assertion
Ref Expression
breprexpnat (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
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.
StepHypRef Expression
1 breprexp.n . . . 4 (𝜑𝑁 ∈ ℕ0)
2 breprexp.s . . . 4 (𝜑𝑆 ∈ ℕ0)
3 breprexp.z . . . 4 (𝜑𝑍 ∈ ℂ)
4 fvex 6919 . . . . . 6 ((𝟭‘ℕ)‘𝐴) ∈ V
54fconst 6794 . . . . 5 ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6 nnex 12272 . . . . . . . . 9 ℕ ∈ V
7 breprexpnat.a . . . . . . . . 9 (𝜑𝐴 ⊆ ℕ)
8 indf 32840 . . . . . . . . 9 ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
96, 7, 8sylancr 587 . . . . . . . 8 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10 0cn 11253 . . . . . . . . 9 0 ∈ ℂ
11 ax-1cn 11213 . . . . . . . . 9 1 ∈ ℂ
12 prssi 4821 . . . . . . . . 9 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
1310, 11, 12mp2an 692 . . . . . . . 8 {0, 1} ⊆ ℂ
14 fss 6752 . . . . . . . 8 ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
159, 13, 14sylancl 586 . . . . . . 7 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16 cnex 11236 . . . . . . . 8 ℂ ∈ V
1716, 6elmap 8911 . . . . . . 7 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
1815, 17sylibr 234 . . . . . 6 (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
194snss 4785 . . . . . 6 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
2018, 19sylib 218 . . . . 5 (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21 fss 6752 . . . . 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))
225, 20, 21sylancr 587 . . . 4 (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
231, 2, 3, 22breprexp 34648 . . 3 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
244fvconst2 7224 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2524ad2antlr 727 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2625fveq1d 6908 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
2726oveq1d 7446 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
2827sumeq2dv 15738 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
296a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ℕ ∈ V)
30 fzfi 14013 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32 fz1ssnn 13595 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3332a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
347adantr 480 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
353ad2antrr 726 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36 nnssnn0 12529 . . . . . . . . . 10 ℕ ⊆ ℕ0
3732, 36sstri 3993 . . . . . . . . 9 (1...𝑁) ⊆ ℕ0
38 simpr 484 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
3937, 38sselid 3981 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
4035, 39expcld 14186 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍𝑏) ∈ ℂ)
4129, 31, 33, 34, 40indsumin 32847 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏))
42 incom 4209 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 15736 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4528, 41, 443eqtrd 2781 . . . . 5 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4645prodeq2dv 15958 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
47 fzofi 14015 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (𝜑 → (0..^𝑆) ∈ Fin)
49 inss2 4238 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50 ssfi 9213 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
5130, 49, 50mp2an 692 . . . . . . 7 (𝐴 ∩ (1...𝑁)) ∈ Fin
5251a1i 11 . . . . . 6 (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
533adantr 480 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
5449, 37sstri 3993 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55 simpr 484 . . . . . . . 8 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sselid 3981 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
5753, 56expcld 14186 . . . . . 6 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍𝑏) ∈ ℂ)
5852, 57fsumcl 15769 . . . . 5 (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ)
59 fprodconst 16014 . . . . 5 (((0..^𝑆) ∈ Fin ∧ Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ) → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))))
6048, 58, 59syl2anc 584 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))))
61 hashfzo0 14469 . . . . . 6 (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
6362oveq2d 7447 . . . 4 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6446, 60, 633eqtrd 2781 . . 3 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6532a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
66 fzssz 13566 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℤ
67 simpr 484 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
6866, 67sselid 3981 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
692adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
7030a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
7165, 68, 69, 70reprfi 34631 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
723adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73 fz0ssnn0 13662 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℕ0
7473, 67sselid 3981 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
7572, 74expcld 14186 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑍𝑚) ∈ ℂ)
7647a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
779ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
7832a1i 11 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
7968adantr 480 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
8069adantr 480 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
8278, 79, 80, 81reprf 34627 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
8382ffvelcdmda 7104 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
8432, 83sselid 3981 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
8577, 84ffvelcdmd 7105 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ {0, 1})
8613, 85sselid 3981 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8776, 86fprodcl 15988 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8871, 75, 87fsummulc1 15821 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) · (𝑍𝑚)))
897adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
9089, 68, 69, 70, 65hashreprin 34635 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9190oveq1d 7446 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) · (𝑍𝑚)))
9224fveq1d 6908 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9392adantl 481 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑎 ∈ (0..^𝑆)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9493prodeq2dv 15958 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9594adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9695oveq1d 7446 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) · (𝑍𝑚)))
9796sumeq2dv 15738 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) · (𝑍𝑚)))
9888, 91, 973eqtr4rd 2788 . . . 4 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚)))
9998sumeq2dv 15738 . . 3 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚)))
10023, 64, 993eqtr3d 2785 . 2 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚)))
101 breprexpnat.p . . 3 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)
102101oveq1i 7441 . 2 (𝑃𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104103oveq1i 7441 . . . 4 (𝑅 · (𝑍𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
105104a1i 11 . . 3 (𝑚 ∈ (0...(𝑆 · 𝑁)) → (𝑅 · (𝑍𝑚)) = ((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚)))
106105sumeq2i 15734 . 2 Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
107100, 102, 1063eqtr4g 2802 1 (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  {csn 4626  {cpr 4628   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   · cmul 11160  cn 12266  0cn0 12526  cz 12613  ...cfz 13547  ..^cfzo 13694  cexp 14102  chash 14369  Σcsu 15722  cprod 15939  𝟭cind 32835  reprcrepr 34623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-prod 15940  df-ind 32836  df-repr 34624
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator