Step |
Hyp |
Ref |
Expression |
1 |
|
2fveq3 |
⊢ ( 𝐴 = 0ℎ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( normfn ‘ ( bra ‘ 0ℎ ) ) ) |
2 |
|
fveq2 |
⊢ ( 𝐴 = 0ℎ → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ 0ℎ ) ) |
3 |
1 2
|
eqeq12d |
⊢ ( 𝐴 = 0ℎ → ( ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ↔ ( normfn ‘ ( bra ‘ 0ℎ ) ) = ( normℎ ‘ 0ℎ ) ) ) |
4 |
|
brafn |
⊢ ( 𝐴 ∈ ℋ → ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ ) |
5 |
|
nmfnval |
⊢ ( ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) ) |
8 |
|
nmfnsetre |
⊢ ( ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ ) |
9 |
4 8
|
syl |
⊢ ( 𝐴 ∈ ℋ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ ) |
10 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
11 |
9 10
|
sstrdi |
⊢ ( 𝐴 ∈ ℋ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ) |
12 |
|
normcl |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℝ ) |
13 |
12
|
rexrd |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℝ* ) |
14 |
11 13
|
jca |
⊢ ( 𝐴 ∈ ℋ → ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( normℎ ‘ 𝐴 ) ∈ ℝ* ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( normℎ ‘ 𝐴 ) ∈ ℝ* ) ) |
16 |
|
vex |
⊢ 𝑧 ∈ V |
17 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑥 = 𝑧 → ( ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) ) |
19 |
18
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) ) |
20 |
16 19
|
elab |
⊢ ( 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ↔ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) |
21 |
|
id |
⊢ ( 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) → 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) |
22 |
|
braval |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) = ( 𝑦 ·ih 𝐴 ) ) |
23 |
22
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) → ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) |
25 |
21 24
|
sylan9eqr |
⊢ ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) |
26 |
|
bcs2 |
⊢ ( ( 𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) |
27 |
26
|
3expa |
⊢ ( ( ( 𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) |
28 |
27
|
ancom1s |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) |
29 |
28
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) |
30 |
25 29
|
eqbrtrd |
⊢ ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( normℎ ‘ 𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) |
31 |
30
|
exp41 |
⊢ ( 𝐴 ∈ ℋ → ( 𝑦 ∈ ℋ → ( ( normℎ ‘ 𝑦 ) ≤ 1 → ( 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) ) ) ) |
32 |
31
|
imp4a |
⊢ ( 𝐴 ∈ ℋ → ( 𝑦 ∈ ℋ → ( ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) ) ) |
33 |
32
|
rexlimdv |
⊢ ( 𝐴 ∈ ℋ → ( ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) ) |
34 |
33
|
imp |
⊢ ( ( 𝐴 ∈ ℋ ∧ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) |
35 |
20 34
|
sylan2b |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ) → 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) |
36 |
35
|
ralrimiva |
⊢ ( 𝐴 ∈ ℋ → ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( normℎ ‘ 𝐴 ) ) |
38 |
12
|
recnd |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℂ ) |
39 |
38
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ∈ ℂ ) |
40 |
|
normne0 |
⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0ℎ ) ) |
41 |
40
|
biimpar |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ≠ 0 ) |
42 |
39 41
|
reccld |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℂ ) |
43 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 𝐴 ∈ ℋ ) |
44 |
|
hvmulcl |
⊢ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ∈ ℋ ) |
45 |
42 43 44
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ∈ ℋ ) |
46 |
|
norm1 |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) = 1 ) |
47 |
|
1le1 |
⊢ 1 ≤ 1 |
48 |
46 47
|
eqbrtrdi |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) ≤ 1 ) |
49 |
|
ax-his3 |
⊢ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
50 |
42 43 43 49
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
51 |
12
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ∈ ℝ ) |
52 |
51 41
|
rereccld |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℝ ) |
53 |
|
hiidrcl |
⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ ) |
54 |
53
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( 𝐴 ·ih 𝐴 ) ∈ ℝ ) |
55 |
52 54
|
remulcld |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ∈ ℝ ) |
56 |
50 55
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ∈ ℝ ) |
57 |
|
normgt0 |
⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ≠ 0ℎ ↔ 0 < ( normℎ ‘ 𝐴 ) ) ) |
58 |
57
|
biimpa |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( normℎ ‘ 𝐴 ) ) |
59 |
51 58
|
recgt0d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( 1 / ( normℎ ‘ 𝐴 ) ) ) |
60 |
|
0re |
⊢ 0 ∈ ℝ |
61 |
|
ltle |
⊢ ( ( 0 ∈ ℝ ∧ ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℝ ) → ( 0 < ( 1 / ( normℎ ‘ 𝐴 ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) ) ) |
62 |
60 61
|
mpan |
⊢ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℝ → ( 0 < ( 1 / ( normℎ ‘ 𝐴 ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) ) ) |
63 |
52 59 62
|
sylc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) ) |
64 |
|
hiidge0 |
⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( 𝐴 ·ih 𝐴 ) ) |
65 |
64
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 ≤ ( 𝐴 ·ih 𝐴 ) ) |
66 |
52 54 63 65
|
mulge0d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 ≤ ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
67 |
66 50
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 ≤ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) |
68 |
56 67
|
absidd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) = ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) |
69 |
39 41
|
recid2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) = 1 ) |
70 |
69
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( normℎ ‘ 𝐴 ) · ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) ) = ( ( normℎ ‘ 𝐴 ) · 1 ) ) |
71 |
39 42 39
|
mul12d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( normℎ ‘ 𝐴 ) · ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( ( normℎ ‘ 𝐴 ) · ( normℎ ‘ 𝐴 ) ) ) ) |
72 |
38
|
sqvald |
⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( ( normℎ ‘ 𝐴 ) · ( normℎ ‘ 𝐴 ) ) ) |
73 |
|
normsq |
⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) ) |
74 |
72 73
|
eqtr3d |
⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) · ( normℎ ‘ 𝐴 ) ) = ( 𝐴 ·ih 𝐴 ) ) |
75 |
74
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( normℎ ‘ 𝐴 ) · ( normℎ ‘ 𝐴 ) ) = ( 𝐴 ·ih 𝐴 ) ) |
76 |
75
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( ( normℎ ‘ 𝐴 ) · ( normℎ ‘ 𝐴 ) ) ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
77 |
71 76
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( normℎ ‘ 𝐴 ) · ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
78 |
38
|
mulid1d |
⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) · 1 ) = ( normℎ ‘ 𝐴 ) ) |
79 |
78
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( normℎ ‘ 𝐴 ) · 1 ) = ( normℎ ‘ 𝐴 ) ) |
80 |
70 77 79
|
3eqtr3rd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ) |
81 |
50 68 80
|
3eqtr4rd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) ) |
82 |
|
fveq2 |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) → ( normℎ ‘ 𝑦 ) = ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) ) |
83 |
82
|
breq1d |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) → ( ( normℎ ‘ 𝑦 ) ≤ 1 ↔ ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) ≤ 1 ) ) |
84 |
|
fvoveq1 |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) = ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) ) |
85 |
84
|
eqeq2d |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) → ( ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ↔ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) ) ) |
86 |
83 85
|
anbi12d |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) → ( ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ↔ ( ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) ) ) ) |
87 |
86
|
rspcev |
⊢ ( ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ∈ ℋ ∧ ( ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ·ih 𝐴 ) ) ) ) → ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) |
88 |
45 48 81 87
|
syl12anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) |
89 |
23
|
eqeq2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) |
90 |
89
|
anbi2d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) ) |
91 |
90
|
rexbidva |
⊢ ( 𝐴 ∈ ℋ → ( ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) ) |
93 |
88 92
|
mpbird |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) |
94 |
|
eqeq1 |
⊢ ( 𝑥 = ( normℎ ‘ 𝐴 ) → ( 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) |
95 |
94
|
anbi2d |
⊢ ( 𝑥 = ( normℎ ‘ 𝐴 ) → ( ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) ) |
96 |
95
|
rexbidv |
⊢ ( 𝑥 = ( normℎ ‘ 𝐴 ) → ( ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ ( normℎ ‘ 𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) ) |
97 |
39 93 96
|
elabd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ) |
98 |
|
breq2 |
⊢ ( 𝑤 = ( normℎ ‘ 𝐴 ) → ( 𝑧 < 𝑤 ↔ 𝑧 < ( normℎ ‘ 𝐴 ) ) ) |
99 |
98
|
rspcev |
⊢ ( ( ( normℎ ‘ 𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ∧ 𝑧 < ( normℎ ‘ 𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) |
100 |
97 99
|
sylan |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑧 < ( normℎ ‘ 𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) |
101 |
100
|
adantlr |
⊢ ( ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( normℎ ‘ 𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) |
102 |
101
|
ex |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑧 ∈ ℝ ) → ( 𝑧 < ( normℎ ‘ 𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) ) |
103 |
102
|
ralrimiva |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∀ 𝑧 ∈ ℝ ( 𝑧 < ( normℎ ‘ 𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) ) |
104 |
|
supxr2 |
⊢ ( ( ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( normℎ ‘ 𝐴 ) ∈ ℝ* ) ∧ ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( normℎ ‘ 𝐴 ) ∧ ∀ 𝑧 ∈ ℝ ( 𝑧 < ( normℎ ‘ 𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) ) ) → sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) = ( normℎ ‘ 𝐴 ) ) |
105 |
15 37 103 104
|
syl12anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( normℎ ‘ 𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) = ( normℎ ‘ 𝐴 ) ) |
106 |
7 105
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) |
107 |
|
nmfn0 |
⊢ ( normfn ‘ ( ℋ × { 0 } ) ) = 0 |
108 |
|
bra0 |
⊢ ( bra ‘ 0ℎ ) = ( ℋ × { 0 } ) |
109 |
108
|
fveq2i |
⊢ ( normfn ‘ ( bra ‘ 0ℎ ) ) = ( normfn ‘ ( ℋ × { 0 } ) ) |
110 |
|
norm0 |
⊢ ( normℎ ‘ 0ℎ ) = 0 |
111 |
107 109 110
|
3eqtr4i |
⊢ ( normfn ‘ ( bra ‘ 0ℎ ) ) = ( normℎ ‘ 0ℎ ) |
112 |
111
|
a1i |
⊢ ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 0ℎ ) ) = ( normℎ ‘ 0ℎ ) ) |
113 |
3 106 112
|
pm2.61ne |
⊢ ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) |