Metamath Proof Explorer

Theorem etransclem33

Description: F is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses etransclem33.s ⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
etransclem33.x ⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
etransclem33.p ⊢ ( 𝜑 → 𝑃 ∈ ℕ )
etransclem33.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
etransclem33.f ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) )
etransclem33.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion etransclem33 ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) : 𝑋 ⟶ ℂ )

Proof

Step Hyp Ref Expression
1 etransclem33.s ⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
2 etransclem33.x ⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
3 etransclem33.p ⊢ ( 𝜑 → 𝑃 ∈ ℕ )
4 etransclem33.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
5 etransclem33.f ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) )
6 etransclem33.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7 eqidd ⊢ ( 𝜑 → ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) = ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) )
8 oveq2 ⊢ ( 𝑚 = 𝑁 → ( 0 ... 𝑚 ) = ( 0 ... 𝑁 ) )
9 8 oveq1d ⊢ ( 𝑚 = 𝑁 → ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) = ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) )
10 eqeq2 ⊢ ( 𝑚 = 𝑁 → ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 ↔ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 ) )
11 9 10 rabeqbidv ⊢ ( 𝑚 = 𝑁 → { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } = { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } )
12 11 adantl ⊢ ( ( 𝜑 ∧ 𝑚 = 𝑁 ) → { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } = { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } )
13 ovex ⊢ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∈ V
14 13 rabex ⊢ { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ∈ V
15 14 a1i ⊢ ( 𝜑 → { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ∈ V )
16 7 12 6 15 fvmptd ⊢ ( 𝜑 → ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) = { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } )
17 fzfi ⊢ ( 0 ... 𝑁 ) ∈ Fin
18 fzfi ⊢ ( 0 ... 𝑀 ) ∈ Fin
19 mapfi ⊢ ( ( ( 0 ... 𝑁 ) ∈ Fin ∧ ( 0 ... 𝑀 ) ∈ Fin ) → ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∈ Fin )
20 17 18 19 mp2an ⊢ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∈ Fin
21 ssrab2 ⊢ { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ⊆ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) )
22 ssfi ⊢ ( ( ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∈ Fin ∧ { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ⊆ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ) → { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ∈ Fin )
23 20 21 22 mp2an ⊢ { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } ∈ Fin
24 16 23 eqeltrdi ⊢ ( 𝜑 → ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ∈ Fin )
25 24 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ∈ Fin )
26 6 faccld ⊢ ( 𝜑 → ( ! ‘ 𝑁 ) ∈ ℕ )
27 26 nncnd ⊢ ( 𝜑 → ( ! ‘ 𝑁 ) ∈ ℂ )
28 27 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ( ! ‘ 𝑁 ) ∈ ℂ )
29 18 a1i ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ( 0 ... 𝑀 ) ∈ Fin )
30 simpr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) )
31 16 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) = { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } )
32 30 31 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → 𝑐 ∈ { 𝑑 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑁 } )
33 21 32 sselid ⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → 𝑐 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) )
34 elmapi ⊢ ( 𝑐 ∈ ( ( 0 ... 𝑁 ) ↑m ( 0 ... 𝑀 ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... 𝑁 ) )
35 33 34 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... 𝑁 ) )
36 35 ffvelcdmda ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ( 0 ... 𝑁 ) )
37 36 adantllr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ( 0 ... 𝑁 ) )
38 elfznn0 ⊢ ( ( 𝑐 ‘ 𝑗 ) ∈ ( 0 ... 𝑁 ) → ( 𝑐 ‘ 𝑗 ) ∈ ℕ0 )
39 37 38 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ℕ0 )
40 39 faccld ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℕ )
41 40 nncnd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℂ )
42 29 41 fprodcl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℂ )
43 40 nnne0d ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ≠ 0 )
44 29 41 43 fprodn0 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ≠ 0 )
45 28 42 44 divcld ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ∈ ℂ )
46 1 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑆 ∈ { ℝ , ℂ } )
47 2 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
48 3 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑃 ∈ ℕ )
49 etransclem5 ⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑤 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑧 ∈ 𝑋 ↦ ( ( 𝑧 − 𝑤 ) ↑ if ( 𝑤 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
50 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) )
51 46 47 48 49 50 39 etransclem20 ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) : 𝑋 ⟶ ℂ )
52 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑥 ∈ 𝑋 )
53 51 52 ffvelcdmd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ∈ ℂ )
54 29 53 fprodcl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ∈ ℂ )
55 45 54 mulcld ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ) → ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ∈ ℂ )
56 25 55 fsumcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ∈ ℂ )
57 eqid ⊢ ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) )
58 56 57 fmptd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ) : 𝑋 ⟶ ℂ )
59 etransclem5 ⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
60 etransclem11 ⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = 𝑛 } )
61 1 2 3 4 5 6 59 60 etransclem30 ⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ) )
62 61 feq1d ⊢ ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) : 𝑋 ⟶ ℂ ↔ ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) ‘ 𝑗 ) ) ‘ ( 𝑐 ‘ 𝑗 ) ) ‘ 𝑥 ) ) ) : 𝑋 ⟶ ℂ ) )
63 58 62 mpbird ⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) : 𝑋 ⟶ ℂ )