Step |
Hyp |
Ref |
Expression |
1 |
|
plyadd.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
2 |
|
plyadd.2 |
⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |
3 |
|
plyadd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
4 |
|
plyadd.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
5 |
|
plyadd.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
6 |
|
plyadd.a |
⊢ ( 𝜑 → 𝐴 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) |
7 |
|
plyadd.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) |
8 |
|
plyadd.a2 |
⊢ ( 𝜑 → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
9 |
|
plyadd.b2 |
⊢ ( 𝜑 → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
10 |
|
plyadd.f |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
11 |
|
plyadd.g |
⊢ ( 𝜑 → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
12 |
|
plybss |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑆 ⊆ ℂ ) |
13 |
1 12
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
14 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
15 |
14
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ ℂ ) |
16 |
13 15
|
unssd |
⊢ ( 𝜑 → ( 𝑆 ∪ { 0 } ) ⊆ ℂ ) |
17 |
|
cnex |
⊢ ℂ ∈ V |
18 |
|
ssexg |
⊢ ( ( ( 𝑆 ∪ { 0 } ) ⊆ ℂ ∧ ℂ ∈ V ) → ( 𝑆 ∪ { 0 } ) ∈ V ) |
19 |
16 17 18
|
sylancl |
⊢ ( 𝜑 → ( 𝑆 ∪ { 0 } ) ∈ V ) |
20 |
|
nn0ex |
⊢ ℕ0 ∈ V |
21 |
|
elmapg |
⊢ ( ( ( 𝑆 ∪ { 0 } ) ∈ V ∧ ℕ0 ∈ V ) → ( 𝐴 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
22 |
19 20 21
|
sylancl |
⊢ ( 𝜑 → ( 𝐴 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
23 |
6 22
|
mpbid |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
24 |
23 16
|
fssd |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
25 |
|
elmapg |
⊢ ( ( ( 𝑆 ∪ { 0 } ) ∈ V ∧ ℕ0 ∈ V ) → ( 𝐵 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝐵 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
26 |
19 20 25
|
sylancl |
⊢ ( 𝜑 → ( 𝐵 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝐵 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
27 |
7 26
|
mpbid |
⊢ ( 𝜑 → 𝐵 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
28 |
27 16
|
fssd |
⊢ ( 𝜑 → 𝐵 : ℕ0 ⟶ ℂ ) |
29 |
1 2 4 5 24 28 8 9 10 11
|
plyaddlem1 |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
30 |
5 4
|
ifcld |
⊢ ( 𝜑 → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℕ0 ) |
31 |
|
eqid |
⊢ ( 𝑆 ∪ { 0 } ) = ( 𝑆 ∪ { 0 } ) |
32 |
13 31 3
|
un0addcl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑆 ∪ { 0 } ) ∧ 𝑦 ∈ ( 𝑆 ∪ { 0 } ) ) ) → ( 𝑥 + 𝑦 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
33 |
20
|
a1i |
⊢ ( 𝜑 → ℕ0 ∈ V ) |
34 |
|
inidm |
⊢ ( ℕ0 ∩ ℕ0 ) = ℕ0 |
35 |
32 23 27 33 33 34
|
off |
⊢ ( 𝜑 → ( 𝐴 ∘f + 𝐵 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
36 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
37 |
|
ffvelrn |
⊢ ( ( ( 𝐴 ∘f + 𝐵 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
38 |
35 36 37
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
39 |
16 30 38
|
elplyd |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ∈ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) ) |
40 |
29 39
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) ∈ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) ) |
41 |
|
plyun0 |
⊢ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) = ( Poly ‘ 𝑆 ) |
42 |
40 41
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) ∈ ( Poly ‘ 𝑆 ) ) |