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Theorem breprexpnat 34627
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 12269 . . . . . . . . 9 ℕ ∈ V
7 breprexpnat.a . . . . . . . . 9 (𝜑𝐴 ⊆ ℕ)
8 indf 33995 . . . . . . . . 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 11250 . . . . . . . . 9 0 ∈ ℂ
11 ax-1cn 11210 . . . . . . . . 9 1 ∈ ℂ
12 prssi 4825 . . . . . . . . 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 11233 . . . . . . . 8 ℂ ∈ V
1716, 6elmap 8909 . . . . . . 7 (((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ) ↔ ((𝟭‘ℕ)‘𝐴):ℕ⟶ℂ)
1815, 17sylibr 234 . . . . . 6 (𝜑 → ((𝟭‘ℕ)‘𝐴) ∈ (ℂ ↑m ℕ))
194snss 4789 . . . . . 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 34626 . . 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 7223 . . . . . . . . . 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 7445 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)))
2827sumeq2dv 15734 . . . . . 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 14009 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → (1...𝑁) ∈ Fin)
32 fz1ssnn 13591 . . . . . . . 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 12526 . . . . . . . . . 10 ℕ ⊆ ℕ0
3732, 36sstri 4004 . . . . . . . . 9 (1...𝑁) ⊆ ℕ0
38 simpr 484 . . . . . . . . 9 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
3937, 38sselid 3992 . . . . . . . 8 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℕ0)
4035, 39expcld 14182 . . . . . . 7 (((𝜑𝑎 ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) → (𝑍𝑏) ∈ ℂ)
4129, 31, 33, 34, 40indsumin 34002 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)((((𝟭‘ℕ)‘𝐴)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏))
42 incom 4216 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (0..^𝑆)) → ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 15732 . . . . . 6 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4528, 41, 443eqtrd 2778 . . . . 5 ((𝜑𝑎 ∈ (0..^𝑆)) → Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
4645prodeq2dv 15954 . . . 4 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((((0..^𝑆) × {((𝟭‘ℕ)‘𝐴)})‘𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏))
47 fzofi 14011 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (𝜑 → (0..^𝑆) ∈ Fin)
49 inss2 4245 . . . . . . . 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 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 4004 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) ⊆ ℕ0
55 simpr 484 . . . . . . . 8 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sselid 3992 . . . . . . 7 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → 𝑏 ∈ ℕ0)
5753, 56expcld 14182 . . . . . 6 ((𝜑𝑏 ∈ (𝐴 ∩ (1...𝑁))) → (𝑍𝑏) ∈ ℂ)
5852, 57fsumcl 15765 . . . . 5 (𝜑 → Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏) ∈ ℂ)
59 fprodconst 16010 . . . . 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 14465 . . . . . 6 (𝑆 ∈ ℕ0 → (♯‘(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (𝜑 → (♯‘(0..^𝑆)) = 𝑆)
6362oveq2d 7446 . . . 4 (𝜑 → (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑(♯‘(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆))
6446, 60, 633eqtrd 2778 . . 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 13562 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℤ
67 simpr 484 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
6866, 67sselid 3992 . . . . . . 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 34609 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
723adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑍 ∈ ℂ)
73 fz0ssnn0 13658 . . . . . . . 8 (0...(𝑆 · 𝑁)) ⊆ ℕ0
7473, 67sselid 3992 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℕ0)
7572, 74expcld 14182 . . . . . 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 34605 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
8382ffvelcdmda 7103 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
8432, 83sselid 3992 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
8577, 84ffvelcdmd 7104 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ {0, 1})
8613, 85sselid 3992 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8776, 86fprodcl 15984 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) ∈ ℂ)
8871, 75, 87fsummulc1 15817 . . . . 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 34613 . . . . . 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 7445 . . . . 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 15954 . . . . . . . 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 7445 . . . . . 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 15734 . . . . 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 2785 . . . 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 15734 . . 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 2782 . 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 7440 . 2 (𝑃𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))
104103oveq1i 7440 . . . 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 15730 . 2 Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))((♯‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) · (𝑍𝑚))
107100, 102, 1063eqtr4g 2799 1 (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  wss 3962  {csn 4630  {cpr 4632   × cxp 5686  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  Fincfn 8983  cc 11150  0cc0 11152  1c1 11153   · cmul 11157  cn 12263  0cn0 12523  cz 12610  ...cfz 13543  ..^cfzo 13690  cexp 14098  chash 14365  Σcsu 15718  cprod 15935  𝟭cind 33990  reprcrepr 34601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-prod 15936  df-ind 33991  df-repr 34602
This theorem is referenced by: (None)
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