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Theorem breprexpnat 34480
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 6914 . . . . . 6 ((𝟭‘ℕ)‘𝐴) ∈ V
54fconst 6788 . . . . 5 ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶{((𝟭‘ℕ)‘𝐴)}
6 nnex 12270 . . . . . . . . 9 ℕ ∈ V
7 breprexpnat.a . . . . . . . . 9 (𝜑𝐴 ⊆ ℕ)
8 indf 33848 . . . . . . . . 9 ((ℕ ∈ V ∧ 𝐴 ⊆ ℕ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
96, 7, 8sylancr 585 . . . . . . . 8 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1})
10 0cn 11256 . . . . . . . . 9 0 ∈ ℂ
11 ax-1cn 11216 . . . . . . . . 9 1 ∈ ℂ
12 prssi 4830 . . . . . . . . 9 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
1310, 11, 12mp2an 690 . . . . . . . 8 {0, 1} ⊆ ℂ
14 fss 6744 . . . . . . . 8 ((((𝟭‘ℕ)‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℂ) → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
159, 13, 14sylancl 584 . . . . . . 7 (𝜑 → ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
16 cnex 11239 . . . . . . . 8 ℂ ∈ V
1716, 6elmap 8900 . . . . . . 7 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
1815, 17sylibr 233 . . . . . 6 (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
194snss 4794 . . . . . 6 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
2018, 19sylib 217 . . . . 5 (𝜑 → {((𝟭‘ℕ)‘𝐴)} ⊆ (ℂ ↑m ℕ))
21 fss 6744 . . . . 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 585 . . . 4 (𝜑 → ((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)}):(0..^𝑆)⟶(ℂ ↑m ℕ))
231, 2, 3, 22breprexp 34479 . . 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 7221 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2524ad2antlr 725 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎) = ((𝟭‘ℕ)‘𝐴))
2625fveq1d 6903 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) = (((𝟭‘ℕ)‘𝐴)‘𝑏))
2726oveq1d 7439 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
2827sumeq2dv 15707 . . . . . 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 13992 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32 fz1ssnn 13586 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3332a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ⊆ ℕ)
347adantr 479 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
353ad2antrr 724 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑍 ∈ ℂ)
36 nnssnn0 12527 . . . . . . . . . 10 ℕ ⊆ ℕ0
3732, 36sstri 3989 . . . . . . . . 9 (1...𝑁) ⊆ ℕ0
38 simpr 483 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
3937, 38sselid 3977 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
4035, 39expcld 14165 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍𝑏) ∈ ℂ)
4129, 31, 33, 34, 40indsumin 33855 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏))
42 incom 4202 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 15705 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4528, 41, 443eqtrd 2770 . . . . 5 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4645prodeq2dv 15925 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
47 fzofi 13994 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (𝜑 → (0..^𝑆) ∈ Fin)
49 inss2 4231 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)
50 ssfi 9211 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) → (𝐴 ∩ (1...𝑁)) ∈ Fin)
5130, 49, 50mp2an 690 . . . . . . 7 (𝐴 ∩ (1...𝑁)) ∈ Fin
5251a1i 11 . . . . . 6 (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin)
533adantr 479 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑍 ∈ ℂ)
5449, 37sstri 3989 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55 simpr 483 . . . . . . . 8 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sselid 3977 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
5753, 56expcld 14165 . . . . . 6 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍𝑏) ∈ ℂ)
5852, 57fsumcl 15737 . . . . 5 (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ)
59 fprodconst 15980 . . . . 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 582 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))))
61 hashfzo0 14447 . . . . . 6 (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
6362oveq2d 7440 . . . 4 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6446, 60, 633eqtrd 2770 . . 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 13557 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℤ
67 simpr 483 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
6866, 67sselid 3977 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
692adantr 479 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
7030a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
7165, 68, 69, 70reprfi 34462 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
723adantr 479 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73 fz0ssnn0 13650 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℕ0
7473, 67sselid 3977 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
7572, 74expcld 14165 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑍𝑚) ∈ ℂ)
7647a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
779ad3antrrr 728 . . . . . . . . 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 479 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
8069adantr 479 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
81 simpr 483 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
8278, 79, 80, 81reprf 34458 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
8382ffvelcdmda 7098 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
8432, 83sselid 3977 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
8577, 84ffvelcdmd 7099 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ {0, 1})
8613, 85sselid 3977 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8776, 86fprodcl 15954 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8871, 75, 87fsummulc1 15789 . . . . 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 479 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝐴 ⊆ ℕ)
9089, 68, 69, 70, 65hashreprin 34466 . . . . . 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 7439 . . . . 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 6903 . . . . . . . . . 10 (𝑎 ∈ (0..^𝑆) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9392adantl 480 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑎 ∈ (0..^𝑆)) → ((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9493prodeq2dv 15925 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∏𝑎 ∈ (0..^𝑆)((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
9594adantr 479 . . . . . . 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 7439 . . . . . 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 15707 . . . . 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 2777 . . . 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 15707 . . 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 2774 . 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 7434 . 2 (𝑃𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104103oveq1i 7434 . . . 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 15703 . 2 Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
107100, 102, 1063eqtr4g 2791 1 (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  cin 3946  wss 3947  {csn 4633  {cpr 4635   × cxp 5680  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  Fincfn 8974  cc 11156  0cc0 11158  1c1 11159   · cmul 11163  cn 12264  0cn0 12524  cz 12610  ...cfz 13538  ..^cfzo 13681  cexp 14081  chash 14347  Σcsu 15690  cprod 15907  𝟭cind 33843  reprcrepr 34454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-disj 5119  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-ico 13384  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691  df-prod 15908  df-ind 33844  df-repr 34455
This theorem is referenced by: (None)
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