Step |
Hyp |
Ref |
Expression |
1 |
|
ulmcl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) → 𝐺 : 𝑆 ⟶ ℂ ) |
3 |
2
|
ffnd |
⊢ ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) → 𝐺 Fn 𝑆 ) |
4 |
|
ulmcl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 → 𝐻 : 𝑆 ⟶ ℂ ) |
5 |
4
|
adantl |
⊢ ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) → 𝐻 : 𝑆 ⟶ ℂ ) |
6 |
5
|
ffnd |
⊢ ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) → 𝐻 Fn 𝑆 ) |
7 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑛 ) |
8 |
|
simplr |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → 𝑛 ∈ ℤ ) |
9 |
|
simpr |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) |
10 |
|
simpllr |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → 𝑥 ∈ 𝑆 ) |
11 |
|
fvex |
⊢ ( ℤ≥ ‘ 𝑛 ) ∈ V |
12 |
11
|
mptex |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ∈ V |
13 |
12
|
a1i |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ∈ V ) |
14 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑘 ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑥 ) ) |
16 |
|
eqid |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) = ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) |
17 |
|
fvex |
⊢ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑥 ) ∈ V |
18 |
15 16 17
|
fvmpt |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑥 ) ) |
19 |
18
|
eqcomd |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑥 ) = ( ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ‘ 𝑘 ) ) |
20 |
19
|
adantl |
⊢ ( ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑥 ) = ( ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ‘ 𝑘 ) ) |
21 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) |
22 |
7 8 9 10 13 20 21
|
ulmclm |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ⇝ ( 𝐺 ‘ 𝑥 ) ) |
23 |
|
simp-4r |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) |
24 |
7 8 9 10 13 20 23
|
ulmclm |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ⇝ ( 𝐻 ‘ 𝑥 ) ) |
25 |
|
climuni |
⊢ ( ( ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ⇝ ( 𝐺 ‘ 𝑥 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ) ⇝ ( 𝐻 ‘ 𝑥 ) ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) |
26 |
22 24 25
|
syl2anc |
⊢ ( ( ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) ∧ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) |
27 |
|
ulmf |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → ∃ 𝑛 ∈ ℤ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) |
28 |
27
|
ad2antrr |
⊢ ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) → ∃ 𝑛 ∈ ℤ 𝐹 : ( ℤ≥ ‘ 𝑛 ) ⟶ ( ℂ ↑m 𝑆 ) ) |
29 |
26 28
|
r19.29a |
⊢ ( ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) |
30 |
3 6 29
|
eqfnfvd |
⊢ ( ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐻 ) → 𝐺 = 𝐻 ) |