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Theorem breprexpnat 31912
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 6656 . . . . . 6 ((𝟭‘ℕ)‘𝐴) ∈ V
54fconst 6538 . . . . 5 ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6 nnex 11621 . . . . . . . . 9 ℕ ∈ V
7 breprexpnat.a . . . . . . . . 9 (𝜑𝐴 ⊆ ℕ)
8 indf 31281 . . . . . . . . 9 ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
96, 7, 8sylancr 590 . . . . . . . 8 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10 0cn 10610 . . . . . . . . 9 0 ∈ ℂ
11 ax-1cn 10572 . . . . . . . . 9 1 ∈ ℂ
12 prssi 4727 . . . . . . . . 9 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
1310, 11, 12mp2an 691 . . . . . . . 8 {0, 1} ⊆ ℂ
14 fss 6500 . . . . . . . 8 ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
159, 13, 14sylancl 589 . . . . . . 7 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16 cnex 10595 . . . . . . . 8 ℂ ∈ V
1716, 6elmap 8410 . . . . . . 7 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
1815, 17sylibr 237 . . . . . 6 (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
194snss 4691 . . . . . 6 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
2018, 19sylib 221 . . . . 5 (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21 fss 6500 . . . . 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 590 . . . 4 (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
231, 2, 3, 22breprexp 31911 . . 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 6939 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2524ad2antlr 726 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2625fveq1d 6645 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
2726oveq1d 7145 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
2827sumeq2dv 15039 . . . . . 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 13323 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32 fz1ssnn 12921 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3332a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
347adantr 484 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
353ad2antrr 725 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36 nnssnn0 11878 . . . . . . . . . 10 ℕ ⊆ ℕ0
3732, 36sstri 3952 . . . . . . . . 9 (1...𝑁) ⊆ ℕ0
38 simpr 488 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
3937, 38sseldi 3941 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
4035, 39expcld 13494 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍𝑏) ∈ ℂ)
4129, 31, 33, 34, 40indsumin 31288 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏))
42 incom 4153 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 15037 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4528, 41, 443eqtrd 2860 . . . . 5 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4645prodeq2dv 15256 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
47 fzofi 13325 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (𝜑 → (0..^𝑆) ∈ Fin)
49 inss2 4181 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50 ssfi 8714 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
5130, 49, 50mp2an 691 . . . . . . 7 (𝐴 ∩ (1...𝑁)) ∈ Fin
5251a1i 11 . . . . . 6 (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
533adantr 484 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
5449, 37sstri 3952 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55 simpr 488 . . . . . . . 8 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sseldi 3941 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
5753, 56expcld 13494 . . . . . 6 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍𝑏) ∈ ℂ)
5852, 57fsumcl 15069 . . . . 5 (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ)
59 fprodconst 15311 . . . . 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 587 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))))
61 hashfzo0 13775 . . . . . 6 (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
6362oveq2d 7146 . . . 4 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6446, 60, 633eqtrd 2860 . . 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 12892 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℤ
67 simpr 488 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
6866, 67sseldi 3941 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
692adantr 484 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
7030a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
7165, 68, 69, 70reprfi 31894 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
723adantr 484 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73 fz0ssnn0 12985 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℕ0
7473, 67sseldi 3941 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
7572, 74expcld 13494 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑍𝑚) ∈ ℂ)
7647a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
779ad3antrrr 729 . . . . . . . . 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 484 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
8069adantr 484 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
8278, 79, 80, 81reprf 31890 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
8382ffvelrnda 6824 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
8432, 83sseldi 3941 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
8577, 84ffvelrnd 6825 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ {0, 1})
8613, 85sseldi 3941 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8776, 86fprodcl 15285 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8871, 75, 87fsummulc1 15119 . . . . 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 484 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
9089, 68, 69, 70, 65hashreprin 31898 . . . . . 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 7145 . . . . 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 6645 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9392adantl 485 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑎 ∈ (0..^𝑆)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9493prodeq2dv 15256 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9594adantr 484 . . . . . . 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 7145 . . . . . 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 15039 . . . . 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 2867 . . . 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 15039 . . 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 2864 . 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 7140 . 2 (𝑃𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104103oveq1i 7140 . . . 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 15035 . 2 Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
107100, 102, 1063eqtr4g 2881 1 (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cin 3909  wss 3910  {csn 4540  {cpr 4542   × cxp 5526  wf 6324  cfv 6328  (class class class)co 7130  m cmap 8381  Fincfn 8484  cc 10512  0cc0 10514  1c1 10515   · cmul 10519  cn 11615  0cn0 11875  cz 11959  ...cfz 12875  ..^cfzo 13016  cexp 13413  chash 13674  Σcsu 15021  cprod 15238  𝟭cind 31276  reprcrepr 31886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-disj 5005  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-sup 8882  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-z 11960  df-uz 12222  df-rp 12368  df-ico 12722  df-fz 12876  df-fzo 13017  df-seq 13353  df-exp 13414  df-hash 13675  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-clim 14824  df-sum 15022  df-prod 15239  df-ind 31277  df-repr 31887
This theorem is referenced by: (None)
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