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Theorem breprexpnat 30942
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 6314 . . . . . 6 ((𝟭‘ℕ)‘𝐴) ∈ V
54fconst 6204 . . . . 5 ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6 nnex 11139 . . . . . . . . 9 ℕ ∈ V
7 breprexpnat.a . . . . . . . . 9 (𝜑𝐴 ⊆ ℕ)
8 indf 30307 . . . . . . . . 9 ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
96, 7, 8sylancr 698 . . . . . . . 8 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10 0cn 10145 . . . . . . . . 9 0 ∈ ℂ
11 ax-1cn 10107 . . . . . . . . 9 1 ∈ ℂ
12 prssi 4461 . . . . . . . . 9 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
1310, 11, 12mp2an 710 . . . . . . . 8 {0, 1} ⊆ ℂ
14 fss 6169 . . . . . . . 8 ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
159, 13, 14sylancl 697 . . . . . . 7 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16 cnex 10130 . . . . . . . 8 ℂ ∈ V
1716, 6elmap 8003 . . . . . . 7 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑𝑚 ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
1815, 17sylibr 224 . . . . . 6 (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑𝑚 ℕ))
194snss 4423 . . . . . 6 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑𝑚 ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑𝑚 ℕ))
2018, 19sylib 208 . . . . 5 (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑𝑚 ℕ))
21 fss 6169 . . . . 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 698 . . . 4 (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
231, 2, 3, 22breprexp 30941 . . 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 6585 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2524ad2antlr 765 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2625fveq1d 6306 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
2726oveq1d 6780 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
2827sumeq2dv 14553 . . . . . 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 12886 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32 fz1ssnn 12486 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3332a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
347adantr 472 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
353ad2antrr 764 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36 nnssnn0 11408 . . . . . . . . . 10 ℕ ⊆ ℕ0
3732, 36sstri 3718 . . . . . . . . 9 (1...𝑁) ⊆ ℕ0
38 simpr 479 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
3937, 38sseldi 3707 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
4035, 39expcld 13123 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍𝑏) ∈ ℂ)
4129, 31, 33, 34, 40indsumin 30314 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏))
42 incom 3913 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 14551 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4528, 41, 443eqtrd 2762 . . . . 5 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4645prodeq2dv 14773 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
47 fzofi 12888 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (𝜑 → (0..^𝑆) ∈ Fin)
49 inss2 3942 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50 ssfi 8296 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
5130, 49, 50mp2an 710 . . . . . . 7 (𝐴 ∩ (1...𝑁)) ∈ Fin
5251a1i 11 . . . . . 6 (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
533adantr 472 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
5449, 37sstri 3718 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55 simpr 479 . . . . . . . 8 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sseldi 3707 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
5753, 56expcld 13123 . . . . . 6 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍𝑏) ∈ ℂ)
5852, 57fsumcl 14584 . . . . 5 (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ)
59 fprodconst 14828 . . . . 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 696 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))))
61 hashfzo0 13330 . . . . . 6 (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
6362oveq2d 6781 . . . 4 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6446, 60, 633eqtrd 2762 . . 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 12457 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℤ
67 simpr 479 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
6866, 67sseldi 3707 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
692adantr 472 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
7030a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
7165, 68, 69, 70reprfi 30924 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
723adantr 472 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73 fz0ssnn0 12549 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℕ0
7473, 67sseldi 3707 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
7572, 74expcld 13123 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑍𝑚) ∈ ℂ)
7647a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
779ad3antrrr 768 . . . . . . . . 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 472 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
8069adantr 472 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81 simpr 479 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
8278, 79, 80, 81reprf 30920 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
8382ffvelrnda 6474 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
8432, 83sseldi 3707 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
8577, 84ffvelrnd 6475 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ {0, 1})
8613, 85sseldi 3707 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8776, 86fprodcl 14802 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8871, 75, 87fsummulc1 14637 . . . . 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 472 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
9089, 68, 69, 70, 65hashreprin 30928 . . . . . 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 6780 . . . . 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 6306 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9392adantl 473 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑎 ∈ (0..^𝑆)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9493prodeq2dv 14773 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9594adantr 472 . . . . . . 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 6780 . . . . . 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 14553 . . . . 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 2769 . . . 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 14553 . . 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 2766 . 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 6775 . 2 (𝑃𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104103oveq1i 6775 . . . 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 14549 . 2 Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
107100, 102, 1063eqtr4g 2783 1 (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  cin 3679  wss 3680  {csn 4285  {cpr 4287   × cxp 5216  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  Fincfn 8072  cc 10047  0cc0 10049  1c1 10050   · cmul 10054  cn 11133  0cn0 11405  cz 11490  ...cfz 12440  ..^cfzo 12580  cexp 12975  chash 13232  Σcsu 14536  cprod 14755  𝟭cind 30302  reprcrepr 30916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-disj 4729  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-ico 12295  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-sum 14537  df-prod 14756  df-ind 30303  df-repr 30917
This theorem is referenced by: (None)
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