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Metamath Proof Explorer

Theorem norm1

Description: From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion norm1 ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) = 1 )

Proof

Step Hyp Ref Expression
1 normcl ⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℝ )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ∈ ℝ )
3 normne0 ⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0ℎ ) )
4 3 biimpar ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ≠ 0 )
5 2 4 rereccld ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℝ )
6 5 recnd ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℂ )
7 simpl ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 𝐴 ∈ ℋ )
8 norm-iii ⊢ ( ( ( 1 / ( normℎ ‘ 𝐴 ) ) ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) = ( ( abs ‘ ( 1 / ( normℎ ‘ 𝐴 ) ) ) · ( normℎ ‘ 𝐴 ) ) )
9 6 7 8 syl2anc ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) = ( ( abs ‘ ( 1 / ( normℎ ‘ 𝐴 ) ) ) · ( normℎ ‘ 𝐴 ) ) )
10 normgt0 ⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ≠ 0ℎ ↔ 0 < ( normℎ ‘ 𝐴 ) ) )
11 10 biimpa ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( normℎ ‘ 𝐴 ) )
12 1re ⊢ 1 ∈ ℝ
13 0le1 ⊢ 0 ≤ 1
14 divge0 ⊢ ( ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ∧ ( ( normℎ ‘ 𝐴 ) ∈ ℝ ∧ 0 < ( normℎ ‘ 𝐴 ) ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) )
15 12 13 14 mpanl12 ⊢ ( ( ( normℎ ‘ 𝐴 ) ∈ ℝ ∧ 0 < ( normℎ ‘ 𝐴 ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) )
16 2 11 15 syl2anc ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 ≤ ( 1 / ( normℎ ‘ 𝐴 ) ) )
17 5 16 absidd ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( abs ‘ ( 1 / ( normℎ ‘ 𝐴 ) ) ) = ( 1 / ( normℎ ‘ 𝐴 ) ) )
18 17 oveq1d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( abs ‘ ( 1 / ( normℎ ‘ 𝐴 ) ) ) · ( normℎ ‘ 𝐴 ) ) = ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) )
19 1 recnd ⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℂ )
20 19 adantr ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ 𝐴 ) ∈ ℂ )
21 20 4 recid2d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( 1 / ( normℎ ‘ 𝐴 ) ) · ( normℎ ‘ 𝐴 ) ) = 1 )
22 9 18 21 3eqtrd ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝐴 ) ) ·ℎ 𝐴 ) ) = 1 )