Step |
Hyp |
Ref |
Expression |
1 |
|
norm1ex.1 |
⊢ 𝐻 ∈ Sℋ |
2 |
|
neeq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ≠ 0ℎ ↔ 𝑧 ≠ 0ℎ ) ) |
3 |
2
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ 𝐻 𝑥 ≠ 0ℎ ↔ ∃ 𝑧 ∈ 𝐻 𝑧 ≠ 0ℎ ) |
4 |
1
|
sheli |
⊢ ( 𝑧 ∈ 𝐻 → 𝑧 ∈ ℋ ) |
5 |
|
normcl |
⊢ ( 𝑧 ∈ ℋ → ( normℎ ‘ 𝑧 ) ∈ ℝ ) |
6 |
4 5
|
syl |
⊢ ( 𝑧 ∈ 𝐻 → ( normℎ ‘ 𝑧 ) ∈ ℝ ) |
7 |
6
|
adantr |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( normℎ ‘ 𝑧 ) ∈ ℝ ) |
8 |
|
normne0 |
⊢ ( 𝑧 ∈ ℋ → ( ( normℎ ‘ 𝑧 ) ≠ 0 ↔ 𝑧 ≠ 0ℎ ) ) |
9 |
4 8
|
syl |
⊢ ( 𝑧 ∈ 𝐻 → ( ( normℎ ‘ 𝑧 ) ≠ 0 ↔ 𝑧 ≠ 0ℎ ) ) |
10 |
9
|
biimpar |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( normℎ ‘ 𝑧 ) ≠ 0 ) |
11 |
7 10
|
rereccld |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝑧 ) ) ∈ ℝ ) |
12 |
11
|
recnd |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝑧 ) ) ∈ ℂ ) |
13 |
|
simpl |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → 𝑧 ∈ 𝐻 ) |
14 |
|
shmulcl |
⊢ ( ( 𝐻 ∈ Sℋ ∧ ( 1 / ( normℎ ‘ 𝑧 ) ) ∈ ℂ ∧ 𝑧 ∈ 𝐻 ) → ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ∈ 𝐻 ) |
15 |
1 14
|
mp3an1 |
⊢ ( ( ( 1 / ( normℎ ‘ 𝑧 ) ) ∈ ℂ ∧ 𝑧 ∈ 𝐻 ) → ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ∈ 𝐻 ) |
16 |
12 13 15
|
syl2anc |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ∈ 𝐻 ) |
17 |
|
norm1 |
⊢ ( ( 𝑧 ∈ ℋ ∧ 𝑧 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ) = 1 ) |
18 |
4 17
|
sylan |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ) = 1 ) |
19 |
|
fveqeq2 |
⊢ ( 𝑦 = ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) → ( ( normℎ ‘ 𝑦 ) = 1 ↔ ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ) = 1 ) ) |
20 |
19
|
rspcev |
⊢ ( ( ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ∈ 𝐻 ∧ ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑧 ) ) ·ℎ 𝑧 ) ) = 1 ) → ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 ) |
21 |
16 18 20
|
syl2anc |
⊢ ( ( 𝑧 ∈ 𝐻 ∧ 𝑧 ≠ 0ℎ ) → ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 ) |
22 |
21
|
rexlimiva |
⊢ ( ∃ 𝑧 ∈ 𝐻 𝑧 ≠ 0ℎ → ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 ) |
23 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
24 |
23
|
neii |
⊢ ¬ 1 = 0 |
25 |
|
eqeq1 |
⊢ ( ( normℎ ‘ 𝑦 ) = 1 → ( ( normℎ ‘ 𝑦 ) = 0 ↔ 1 = 0 ) ) |
26 |
24 25
|
mtbiri |
⊢ ( ( normℎ ‘ 𝑦 ) = 1 → ¬ ( normℎ ‘ 𝑦 ) = 0 ) |
27 |
1
|
sheli |
⊢ ( 𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ ) |
28 |
|
norm-i |
⊢ ( 𝑦 ∈ ℋ → ( ( normℎ ‘ 𝑦 ) = 0 ↔ 𝑦 = 0ℎ ) ) |
29 |
27 28
|
syl |
⊢ ( 𝑦 ∈ 𝐻 → ( ( normℎ ‘ 𝑦 ) = 0 ↔ 𝑦 = 0ℎ ) ) |
30 |
29
|
necon3bbid |
⊢ ( 𝑦 ∈ 𝐻 → ( ¬ ( normℎ ‘ 𝑦 ) = 0 ↔ 𝑦 ≠ 0ℎ ) ) |
31 |
26 30
|
syl5ib |
⊢ ( 𝑦 ∈ 𝐻 → ( ( normℎ ‘ 𝑦 ) = 1 → 𝑦 ≠ 0ℎ ) ) |
32 |
31
|
reximia |
⊢ ( ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 → ∃ 𝑦 ∈ 𝐻 𝑦 ≠ 0ℎ ) |
33 |
|
neeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ≠ 0ℎ ↔ 𝑧 ≠ 0ℎ ) ) |
34 |
33
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐻 𝑦 ≠ 0ℎ ↔ ∃ 𝑧 ∈ 𝐻 𝑧 ≠ 0ℎ ) |
35 |
32 34
|
sylib |
⊢ ( ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 → ∃ 𝑧 ∈ 𝐻 𝑧 ≠ 0ℎ ) |
36 |
22 35
|
impbii |
⊢ ( ∃ 𝑧 ∈ 𝐻 𝑧 ≠ 0ℎ ↔ ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 ) |
37 |
3 36
|
bitri |
⊢ ( ∃ 𝑥 ∈ 𝐻 𝑥 ≠ 0ℎ ↔ ∃ 𝑦 ∈ 𝐻 ( normℎ ‘ 𝑦 ) = 1 ) |