Step |
Hyp |
Ref |
Expression |
1 |
|
occl.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℋ ) |
2 |
|
occl.2 |
⊢ ( 𝜑 → 𝐹 ∈ Cauchy ) |
3 |
|
occl.3 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ⊥ ‘ 𝐴 ) ) |
4 |
|
occl.4 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
5 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
6 |
5
|
cnfldhaus |
⊢ ( TopOpen ‘ ℂfld ) ∈ Haus |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ Haus ) |
8 |
|
ax-hcompl |
⊢ ( 𝐹 ∈ Cauchy → ∃ 𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ) |
9 |
|
hlimf |
⊢ ⇝𝑣 : dom ⇝𝑣 ⟶ ℋ |
10 |
|
ffn |
⊢ ( ⇝𝑣 : dom ⇝𝑣 ⟶ ℋ → ⇝𝑣 Fn dom ⇝𝑣 ) |
11 |
9 10
|
ax-mp |
⊢ ⇝𝑣 Fn dom ⇝𝑣 |
12 |
|
fnbr |
⊢ ( ( ⇝𝑣 Fn dom ⇝𝑣 ∧ 𝐹 ⇝𝑣 𝑥 ) → 𝐹 ∈ dom ⇝𝑣 ) |
13 |
11 12
|
mpan |
⊢ ( 𝐹 ⇝𝑣 𝑥 → 𝐹 ∈ dom ⇝𝑣 ) |
14 |
13
|
rexlimivw |
⊢ ( ∃ 𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 → 𝐹 ∈ dom ⇝𝑣 ) |
15 |
2 8 14
|
3syl |
⊢ ( 𝜑 → 𝐹 ∈ dom ⇝𝑣 ) |
16 |
|
ffun |
⊢ ( ⇝𝑣 : dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) |
17 |
|
funfvbrb |
⊢ ( Fun ⇝𝑣 → ( 𝐹 ∈ dom ⇝𝑣 ↔ 𝐹 ⇝𝑣 ( ⇝𝑣 ‘ 𝐹 ) ) ) |
18 |
9 16 17
|
mp2b |
⊢ ( 𝐹 ∈ dom ⇝𝑣 ↔ 𝐹 ⇝𝑣 ( ⇝𝑣 ‘ 𝐹 ) ) |
19 |
15 18
|
sylib |
⊢ ( 𝜑 → 𝐹 ⇝𝑣 ( ⇝𝑣 ‘ 𝐹 ) ) |
20 |
|
eqid |
⊢ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 |
21 |
|
eqid |
⊢ ( normℎ ∘ −ℎ ) = ( normℎ ∘ −ℎ ) |
22 |
20 21
|
hhims |
⊢ ( normℎ ∘ −ℎ ) = ( IndMet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
23 |
|
eqid |
⊢ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) = ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) |
24 |
20 22 23
|
hhlm |
⊢ ⇝𝑣 = ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ↾ ( ℋ ↑m ℕ ) ) |
25 |
|
resss |
⊢ ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ↾ ( ℋ ↑m ℕ ) ) ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) |
26 |
24 25
|
eqsstri |
⊢ ⇝𝑣 ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) |
27 |
26
|
ssbri |
⊢ ( 𝐹 ⇝𝑣 ( ⇝𝑣 ‘ 𝐹 ) → 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ( ⇝𝑣 ‘ 𝐹 ) ) |
28 |
19 27
|
syl |
⊢ ( 𝜑 → 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ( ⇝𝑣 ‘ 𝐹 ) ) |
29 |
21
|
hilxmet |
⊢ ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) |
30 |
23
|
mopntopon |
⊢ ( ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) → ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ∈ ( TopOn ‘ ℋ ) ) |
31 |
29 30
|
mp1i |
⊢ ( 𝜑 → ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ∈ ( TopOn ‘ ℋ ) ) |
32 |
31
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ ℋ ↦ 𝑥 ) ∈ ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) Cn ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ) |
33 |
1 4
|
sseldd |
⊢ ( 𝜑 → 𝐵 ∈ ℋ ) |
34 |
31 31 33
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ℋ ↦ 𝐵 ) ∈ ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) Cn ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ) |
35 |
20
|
hhnv |
⊢ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ∈ NrmCVec |
36 |
20
|
hhip |
⊢ ·ih = ( ·𝑖OLD ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
37 |
36 22 23 5
|
dipcn |
⊢ ( 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ∈ NrmCVec → ·ih ∈ ( ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ×t ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
38 |
35 37
|
mp1i |
⊢ ( 𝜑 → ·ih ∈ ( ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ×t ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
39 |
31 32 34 38
|
cnmpt12f |
⊢ ( 𝜑 → ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∈ ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
40 |
28 39
|
lmcn |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( ⇝𝑣 ‘ 𝐹 ) ) ) |
41 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ⊥ ‘ 𝐴 ) ) |
42 |
|
ocel |
⊢ ( 𝐴 ⊆ ℋ → ( ( 𝐹 ‘ 𝑘 ) ∈ ( ⊥ ‘ 𝐴 ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ) ) ) |
43 |
1 42
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑘 ) ∈ ( ⊥ ‘ 𝐴 ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ) ) ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( ⊥ ‘ 𝐴 ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ) ) ) |
45 |
41 44
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ) ) |
46 |
45
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ℋ ) |
47 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑘 ) → ( 𝑥 ·ih 𝐵 ) = ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) ) |
48 |
|
eqid |
⊢ ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) = ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) |
49 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) ∈ V |
50 |
47 48 49
|
fvmpt |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ℋ → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) ) |
51 |
46 50
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) ) |
52 |
|
oveq2 |
⊢ ( 𝑥 = 𝐵 → ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) ) |
53 |
52
|
eqeq1d |
⊢ ( 𝑥 = 𝐵 → ( ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ↔ ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) = 0 ) ) |
54 |
45
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑘 ) ·ih 𝑥 ) = 0 ) |
55 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐵 ∈ 𝐴 ) |
56 |
53 54 55
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) ·ih 𝐵 ) = 0 ) |
57 |
51 56
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) = 0 ) |
58 |
|
ocss |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) |
59 |
1 58
|
syl |
⊢ ( 𝜑 → ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) |
60 |
3 59
|
fssd |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ℋ ) |
61 |
|
fvco3 |
⊢ ( ( 𝐹 : ℕ ⟶ ℋ ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
62 |
60 61
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
63 |
|
c0ex |
⊢ 0 ∈ V |
64 |
63
|
fvconst2 |
⊢ ( 𝑘 ∈ ℕ → ( ( ℕ × { 0 } ) ‘ 𝑘 ) = 0 ) |
65 |
64
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℕ × { 0 } ) ‘ 𝑘 ) = 0 ) |
66 |
57 62 65
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( ℕ × { 0 } ) ‘ 𝑘 ) ) |
67 |
66
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( ℕ × { 0 } ) ‘ 𝑘 ) ) |
68 |
|
ovex |
⊢ ( 𝑥 ·ih 𝐵 ) ∈ V |
69 |
68 48
|
fnmpti |
⊢ ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) Fn ℋ |
70 |
|
fnfco |
⊢ ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) Fn ℋ ∧ 𝐹 : ℕ ⟶ ℋ ) → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) Fn ℕ ) |
71 |
69 60 70
|
sylancr |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) Fn ℕ ) |
72 |
63
|
fconst |
⊢ ( ℕ × { 0 } ) : ℕ ⟶ { 0 } |
73 |
|
ffn |
⊢ ( ( ℕ × { 0 } ) : ℕ ⟶ { 0 } → ( ℕ × { 0 } ) Fn ℕ ) |
74 |
72 73
|
ax-mp |
⊢ ( ℕ × { 0 } ) Fn ℕ |
75 |
|
eqfnfv |
⊢ ( ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) Fn ℕ ∧ ( ℕ × { 0 } ) Fn ℕ ) → ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) = ( ℕ × { 0 } ) ↔ ∀ 𝑘 ∈ ℕ ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( ℕ × { 0 } ) ‘ 𝑘 ) ) ) |
76 |
71 74 75
|
sylancl |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) = ( ℕ × { 0 } ) ↔ ∀ 𝑘 ∈ ℕ ( ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( ℕ × { 0 } ) ‘ 𝑘 ) ) ) |
77 |
67 76
|
mpbird |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ∘ 𝐹 ) = ( ℕ × { 0 } ) ) |
78 |
|
fvex |
⊢ ( ⇝𝑣 ‘ 𝐹 ) ∈ V |
79 |
78
|
hlimveci |
⊢ ( 𝐹 ⇝𝑣 ( ⇝𝑣 ‘ 𝐹 ) → ( ⇝𝑣 ‘ 𝐹 ) ∈ ℋ ) |
80 |
|
oveq1 |
⊢ ( 𝑥 = ( ⇝𝑣 ‘ 𝐹 ) → ( 𝑥 ·ih 𝐵 ) = ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) ) |
81 |
|
ovex |
⊢ ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) ∈ V |
82 |
80 48 81
|
fvmpt |
⊢ ( ( ⇝𝑣 ‘ 𝐹 ) ∈ ℋ → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( ⇝𝑣 ‘ 𝐹 ) ) = ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) ) |
83 |
19 79 82
|
3syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐵 ) ) ‘ ( ⇝𝑣 ‘ 𝐹 ) ) = ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) ) |
84 |
40 77 83
|
3brtr3d |
⊢ ( 𝜑 → ( ℕ × { 0 } ) ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) ) |
85 |
5
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
86 |
85
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) |
87 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
88 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
89 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
90 |
89
|
lmconst |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ ) → ( ℕ × { 0 } ) ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) 0 ) |
91 |
86 87 88 90
|
syl3anc |
⊢ ( 𝜑 → ( ℕ × { 0 } ) ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) 0 ) |
92 |
7 84 91
|
lmmo |
⊢ ( 𝜑 → ( ( ⇝𝑣 ‘ 𝐹 ) ·ih 𝐵 ) = 0 ) |