Metamath Proof Explorer


Theorem taylfvallem1

Description: Lemma for taylfval . (Contributed by Mario Carneiro, 30-Dec-2016)

Ref Expression
Hypotheses taylfval.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
taylfval.f ( 𝜑𝐹 : 𝐴 ⟶ ℂ )
taylfval.a ( 𝜑𝐴𝑆 )
taylfval.n ( 𝜑 → ( 𝑁 ∈ ℕ0𝑁 = +∞ ) )
taylfval.b ( ( 𝜑𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) )
Assertion taylfvallem1 ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) · ( ( 𝑋𝐵 ) ↑ 𝑘 ) ) ∈ ℂ )

Proof

Step Hyp Ref Expression
1 taylfval.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
2 taylfval.f ( 𝜑𝐹 : 𝐴 ⟶ ℂ )
3 taylfval.a ( 𝜑𝐴𝑆 )
4 taylfval.n ( 𝜑 → ( 𝑁 ∈ ℕ0𝑁 = +∞ ) )
5 taylfval.b ( ( 𝜑𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) )
6 1 ad2antrr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑆 ∈ { ℝ , ℂ } )
7 cnex ℂ ∈ V
8 7 a1i ( 𝜑 → ℂ ∈ V )
9 elpm2r ( ( ( ℂ ∈ V ∧ 𝑆 ∈ { ℝ , ℂ } ) ∧ ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴𝑆 ) ) → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) )
10 8 1 2 3 9 syl22anc ( 𝜑𝐹 ∈ ( ℂ ↑pm 𝑆 ) )
11 10 ad2antrr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) )
12 simpr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) )
13 12 elin2d ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ℤ )
14 12 elin1d ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ( 0 [,] 𝑁 ) )
15 0xr 0 ∈ ℝ*
16 nn0re ( 𝑁 ∈ ℕ0𝑁 ∈ ℝ )
17 16 rexrd ( 𝑁 ∈ ℕ0𝑁 ∈ ℝ* )
18 id ( 𝑁 = +∞ → 𝑁 = +∞ )
19 pnfxr +∞ ∈ ℝ*
20 18 19 eqeltrdi ( 𝑁 = +∞ → 𝑁 ∈ ℝ* )
21 17 20 jaoi ( ( 𝑁 ∈ ℕ0𝑁 = +∞ ) → 𝑁 ∈ ℝ* )
22 4 21 syl ( 𝜑𝑁 ∈ ℝ* )
23 22 ad2antrr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑁 ∈ ℝ* )
24 elicc1 ( ( 0 ∈ ℝ*𝑁 ∈ ℝ* ) → ( 𝑘 ∈ ( 0 [,] 𝑁 ) ↔ ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘𝑘𝑁 ) ) )
25 15 23 24 sylancr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑘 ∈ ( 0 [,] 𝑁 ) ↔ ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘𝑘𝑁 ) ) )
26 14 25 mpbid ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘𝑘𝑁 ) )
27 26 simp2d ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 0 ≤ 𝑘 )
28 elnn0z ( 𝑘 ∈ ℕ0 ↔ ( 𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ) )
29 13 27 28 sylanbrc ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ℕ0 )
30 dvnf ( ( 𝑆 ∈ { ℝ , ℂ } ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) : dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⟶ ℂ )
31 6 11 29 30 syl3anc ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) : dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⟶ ℂ )
32 5 adantlr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) )
33 31 32 ffvelrnd ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) ∈ ℂ )
34 29 faccld ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ∈ ℕ )
35 34 nncnd ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ∈ ℂ )
36 34 nnne0d ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ≠ 0 )
37 33 35 36 divcld ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ )
38 simplr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑋 ∈ ℂ )
39 2 ad2antrr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐹 : 𝐴 ⟶ ℂ )
40 dvnbss ( ( 𝑆 ∈ { ℝ , ℂ } ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑘 ∈ ℕ0 ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ dom 𝐹 )
41 6 11 29 40 syl3anc ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ dom 𝐹 )
42 39 41 fssdmd ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ 𝐴 )
43 recnprss ( 𝑆 ∈ { ℝ , ℂ } → 𝑆 ⊆ ℂ )
44 1 43 syl ( 𝜑𝑆 ⊆ ℂ )
45 3 44 sstrd ( 𝜑𝐴 ⊆ ℂ )
46 45 ad2antrr ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐴 ⊆ ℂ )
47 42 46 sstrd ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ ℂ )
48 47 32 sseldd ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ ℂ )
49 38 48 subcld ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑋𝐵 ) ∈ ℂ )
50 49 29 expcld ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( 𝑋𝐵 ) ↑ 𝑘 ) ∈ ℂ )
51 37 50 mulcld ( ( ( 𝜑𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) · ( ( 𝑋𝐵 ) ↑ 𝑘 ) ) ∈ ℂ )