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Theorem circlemeth 34486
Description: The Hardy, Littlewood and Ramanujan Circle Method, in a generic form, with different weighting / smoothing functions. (Contributed by Thierry Arnoux, 13-Dec-2021.)
Hypotheses
Ref Expression
circlemeth.n (𝜑𝑁 ∈ ℕ0)
circlemeth.s (𝜑𝑆 ∈ ℕ)
circlemeth.l (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
Assertion
Ref Expression
circlemeth (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
Distinct variable groups:   𝐿,𝑎,𝑐,𝑥   𝑁,𝑎,𝑐,𝑥   𝑆,𝑎,𝑐,𝑥   𝜑,𝑎,𝑐,𝑥

Proof of Theorem circlemeth
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 circlemeth.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
21adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑁 ∈ ℕ0)
3 ioossre 13439 . . . . . . . . 9 (0(,)1) ⊆ ℝ
4 ax-resscn 11215 . . . . . . . . 9 ℝ ⊆ ℂ
53, 4sstri 3989 . . . . . . . 8 (0(,)1) ⊆ ℂ
65a1i 11 . . . . . . 7 (𝜑 → (0(,)1) ⊆ ℂ)
76sselda 3979 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8 circlemeth.s . . . . . . . 8 (𝜑𝑆 ∈ ℕ)
98nnnn0d 12584 . . . . . . 7 (𝜑𝑆 ∈ ℕ0)
109adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑆 ∈ ℕ0)
11 circlemeth.l . . . . . . 7 (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
1211adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
132, 7, 10, 12vtsprod 34485 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → ∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))))
1413oveq1d 7439 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
15 fzfid 13993 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16 ax-icn 11217 . . . . . . . . 9 i ∈ ℂ
17 2cn 12339 . . . . . . . . . 10 2 ∈ ℂ
18 picn 26487 . . . . . . . . . 10 π ∈ ℂ
1917, 18mulcli 11271 . . . . . . . . 9 (2 · π) ∈ ℂ
2016, 19mulcli 11271 . . . . . . . 8 (i · (2 · π)) ∈ ℂ
2120a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
221nn0cnd 12586 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℂ)
2322negcld 11608 . . . . . . . . . 10 (𝜑 → -𝑁 ∈ ℂ)
2423ralrimivw 3140 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
2524r19.21bi 3239 . . . . . . . 8 ((𝜑𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
2625, 7mulcld 11284 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
2721, 26mulcld 11284 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
2827efcld 16085 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29 fz1ssnn 13586 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3029a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31 simpr 483 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
3231elfzelzd 13556 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3332adantlr 713 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3410adantr 479 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
35 fzfid 13993 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
3630, 33, 34, 35reprfi 34462 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37 fzofi 13994 . . . . . . . . 9 (0..^𝑆) ∈ Fin
3837a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
391ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
409ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
4132zcnd 12719 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
4241ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
4311ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
44 simpr 483 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
4529a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
4632adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
479ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
48 simpr 483 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
4945, 46, 47, 48reprf 34458 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
5049ffvelcdmda 7098 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
5129, 50sselid 3977 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
5239, 40, 42, 43, 44, 51breprexplemb 34477 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5352adantl3r 748 . . . . . . . 8 (((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5438, 53fprodcl 15954 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5520a1i 11 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (i · (2 · π)) ∈ ℂ)
5633zcnd 12719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
577adantr 479 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
5856, 57mulcld 11284 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
5955, 58mulcld 11284 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
6059efcld 16085 . . . . . . . 8 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6160adantr 479 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6254, 61mulcld 11284 . . . . . 6 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
6336, 62fsumcl 15737 . . . . 5 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
6415, 28, 63fsummulc1 15789 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
6528adantr 479 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6636, 65, 62fsummulc1 15789 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
6765adantr 479 . . . . . . . . 9 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6854, 61, 67mulassd 11287 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))))
6927adantr 479 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
70 efadd 16096 . . . . . . . . . . . 12 ((((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ ∧ ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
7159, 69, 70syl2anc 582 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
7226adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
7355, 58, 72adddid 11288 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))))
7425adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
7556, 74, 57adddird 11289 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
7622ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
7756, 76negsubd 11627 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚𝑁))
7877oveq1d 7439 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚𝑁) · 𝑥))
7975, 78eqtr3d 2768 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚𝑁) · 𝑥))
8079oveq2d 7440 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8173, 80eqtr3d 2768 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8281fveq2d 6905 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))
8371, 82eqtr3d 2768 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))
8483oveq2d 7440 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8584adantr 479 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8668, 85eqtrd 2766 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8786sumeq2dv 15707 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8866, 87eqtrd 2766 . . . . 5 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8988sumeq2dv 15707 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → Σ𝑚 ∈ (0...(𝑆 · 𝑁))(Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
9014, 64, 893eqtrd 2770 . . 3 ((𝜑𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
9190itgeq2dv 25802 . 2 (𝜑 → ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥 = ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
92 ioombl 25585 . . . . 5 (0(,)1) ∈ dom vol
9392a1i 11 . . . 4 (𝜑 → (0(,)1) ∈ dom vol)
94 fzfid 13993 . . . 4 (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95 sumex 15692 . . . . 5 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ V
9695a1i 11 . . . 4 ((𝜑 ∧ (𝑥 ∈ (0(,)1) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁)))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ V)
9793adantr 479 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
9829a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
999adantr 479 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
100 fzfid 13993 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
10198, 32, 99, 100reprfi 34462 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
10237a1i 11 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
10352adantllr 717 . . . . . . . . 9 (((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
104102, 103fprodcl 15954 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
10556, 76subcld 11621 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
106105, 57mulcld 11284 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) · 𝑥) ∈ ℂ)
10755, 106mulcld 11284 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
108107an32s 650 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
109108adantr 479 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
110109efcld 16085 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
111104, 110mulcld 11284 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ℂ)
112111anasss 465 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ (𝑥 ∈ (0(,)1) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ℂ)
11337a1i 11 . . . . . . . 8 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
114113, 52fprodcl 15954 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
115 fvex 6914 . . . . . . . 8 (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V
116115a1i 11 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
117 ioossicc 13464 . . . . . . . . . 10 (0(,)1) ⊆ (0[,]1)
118117a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ⊆ (0[,]1))
11992a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
120115a1i 11 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0[,]1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
121 0red 11267 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122 1red 11265 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
12322adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
12441, 123subcld 11621 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
125 unitsscn 13531 . . . . . . . . . . . . . 14 (0[,]1) ⊆ ℂ
126125a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127 ssidd 4003 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128 cncfmptc 24923 . . . . . . . . . . . . 13 (((𝑚𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
129124, 126, 127, 128syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
130 cncfmptid 24924 . . . . . . . . . . . . 13 (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
131126, 127, 130syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132129, 131mulcncf 25465 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133132efmul2picn 34442 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134 cniccibl 25861 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ)) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
135121, 122, 133, 134syl3anc 1368 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
136118, 119, 120, 135iblss 25825 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
137136adantr 479 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
138114, 116, 137iblmulc2 25851 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1)
13997, 101, 112, 138itgfsum 25847 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1 ∧ ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥))
140139simpld 493 . . . 4 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1)
14193, 94, 96, 140itgfsum 25847 . . 3 (𝜑 → ((𝑥 ∈ (0(,)1) ↦ Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1 ∧ ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥))
142141simprd 494 . 2 (𝜑 → ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
143 oveq2 7432 . . . . . . 7 (if((𝑚𝑁) = 0, 1, 0) = 1 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 1))
144 oveq2 7432 . . . . . . 7 (if((𝑚𝑁) = 0, 1, 0) = 0 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0))
145101, 114fsumcl 15737 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
146145mulridd 11281 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 1) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
147145mul01d 11463 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0) = 0)
148143, 144, 146, 147ifeq3da 32467 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
149 velsn 4649 . . . . . . . 8 (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
15041, 123subeq0ad 11631 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) = 0 ↔ 𝑚 = 𝑁))
151149, 150bitr4id 289 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚𝑁) = 0))
152151ifbid 4556 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0) = if((𝑚𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
1531nn0zd 12636 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
154153ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
15546, 154zsubcld 12723 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚𝑁) ∈ ℤ)
156 itgexpif 34452 . . . . . . . . . 10 ((𝑚𝑁) ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
157155, 156syl 17 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
158157oveq2d 7440 . . . . . . . 8 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
159158sumeq2dv 15707 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
160 1cnd 11259 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161 0cnd 11257 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162160, 161ifcld 4579 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, 1, 0) ∈ ℂ)
163101, 162, 114fsummulc1 15789 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
164159, 163eqtr4d 2769 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
165148, 152, 1643eqtr4rd 2777 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
166165sumeq2dv 15707 . . . 4 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
167 0zd 12622 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
1689nn0zd 12636 . . . . . . . 8 (𝜑𝑆 ∈ ℤ)
169168, 153zmulcld 12724 . . . . . . 7 (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
1701nn0ge0d 12587 . . . . . . 7 (𝜑 → 0 ≤ 𝑁)
171 nnmulge 32652 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑆 · 𝑁))
1728, 1, 171syl2anc 582 . . . . . . 7 (𝜑𝑁 ≤ (𝑆 · 𝑁))
173167, 169, 153, 170, 172elfzd 13546 . . . . . 6 (𝜑𝑁 ∈ (0...(𝑆 · 𝑁)))
174173snssd 4818 . . . . 5 (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175174sselda 3979 . . . . . . 7 ((𝜑𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
176175, 145syldan 589 . . . . . 6 ((𝜑𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
177176ralrimiva 3136 . . . . 5 (𝜑 → ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
17894olcd 872 . . . . 5 (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179 sumss2 15730 . . . . 5 ((({𝑁} ⊆ (0...(𝑆 · 𝑁)) ∧ ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ) ∧ ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin)) → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
180174, 177, 178, 179syl21anc 836 . . . 4 (𝜑 → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
18129a1i 11 . . . . . . 7 (𝜑 → (1...𝑁) ⊆ ℕ)
182 fzfid 13993 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
183181, 153, 9, 182reprfi 34462 . . . . . 6 (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
18437a1i 11 . . . . . . 7 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
1851ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
1869ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
18722ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
18811ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
189 simpr 483 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
19029a1i 11 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (1...𝑁) ⊆ ℕ)
191153adantr 479 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
1929adantr 479 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ0)
193 simpr 483 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
194190, 191, 192, 193reprf 34458 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195194ffvelcdmda 7098 . . . . . . . . 9 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
19629, 195sselid 3977 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
197185, 186, 187, 188, 189, 196breprexplemb 34477 . . . . . . 7 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
198184, 197fprodcl 15954 . . . . . 6 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
199183, 198fsumcl 15737 . . . . 5 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
200 oveq2 7432 . . . . . . 7 (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201200sumeq1d 15705 . . . . . 6 (𝑚 = 𝑁 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
202201sumsn 15750 . . . . 5 ((𝑁 ∈ ℕ0 ∧ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ) → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
2031, 199, 202syl2anc 582 . . . 4 (𝜑 → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
204166, 180, 2033eqtr2d 2772 . . 3 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
205139simprd 494 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
206110an32s 650 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
207114, 206, 137itgmulc2 25854 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
208207sumeq2dv 15707 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
209205, 208eqtr4d 2769 . . . 4 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥))
210209sumeq2dv 15707 . . 3 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥))
2111, 9reprfz1 34470 . . . 4 (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212211sumeq1d 15705 . . 3 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
213204, 210, 2123eqtr4d 2776 . 2 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
21491, 142, 2133eqtrrd 2771 1 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  wss 3947  ifcif 4533  {csn 4633   class class class wbr 5153  cmpt 5236  dom cdm 5682  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  Fincfn 8974  cc 11156  cr 11157  0cc0 11158  1c1 11159  ici 11160   + caddc 11161   · cmul 11163  cle 11299  cmin 11494  -cneg 11495  cn 12264  2c2 12319  0cn0 12524  cz 12610  cuz 12874  (,)cioo 13378  [,]cicc 13381  ...cfz 13538  ..^cfzo 13681  Σcsu 15690  cprod 15907  expce 16063  πcpi 16068  cnccncf 24887  volcvol 25483  𝐿1cibl 25637  citg 25638  reprcrepr 34454  vtscvts 34481
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-cc 10478  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  ax-addf 11237
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-symdif 4244  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  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-of 7690  df-ofr 7691  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-omul 8501  df-er 8734  df-map 8857  df-pm 8858  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-fi 9454  df-sup 9485  df-inf 9486  df-oi 9553  df-dju 9944  df-card 9982  df-acn 9985  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-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-q 12985  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-ioo 13382  df-ioc 13383  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-fl 13812  df-mod 13890  df-seq 14022  df-exp 14082  df-fac 14291  df-bc 14320  df-hash 14348  df-shft 15072  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-limsup 15473  df-clim 15490  df-rlim 15491  df-sum 15691  df-prod 15908  df-ef 16069  df-sin 16071  df-cos 16072  df-pi 16074  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-starv 17281  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-unif 17289  df-hom 17290  df-cco 17291  df-rest 17437  df-topn 17438  df-0g 17456  df-gsum 17457  df-topgen 17458  df-pt 17459  df-prds 17462  df-xrs 17517  df-qtop 17522  df-imas 17523  df-xps 17525  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-submnd 18774  df-mulg 19062  df-cntz 19311  df-cmn 19780  df-psmet 21335  df-xmet 21336  df-met 21337  df-bl 21338  df-mopn 21339  df-fbas 21340  df-fg 21341  df-cnfld 21344  df-top 22887  df-topon 22904  df-topsp 22926  df-bases 22940  df-cld 23014  df-ntr 23015  df-cls 23016  df-nei 23093  df-lp 23131  df-perf 23132  df-cn 23222  df-cnp 23223  df-haus 23310  df-cmp 23382  df-tx 23557  df-hmeo 23750  df-fil 23841  df-fm 23933  df-flim 23934  df-flf 23935  df-xms 24317  df-ms 24318  df-tms 24319  df-cncf 24889  df-ovol 25484  df-vol 25485  df-mbf 25639  df-itg1 25640  df-itg2 25641  df-ibl 25642  df-itg 25643  df-0p 25690  df-limc 25886  df-dv 25887  df-repr 34455  df-vts 34482
This theorem is referenced by:  circlemethnat  34487  circlevma  34488  circlemethhgt  34489
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