Metamath Proof Explorer


Theorem normsubi

Description: Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999) (New usage is discouraged.)

Ref Expression
Hypotheses normsub.1 𝐴 ∈ ℋ
normsub.2 𝐵 ∈ ℋ
Assertion normsubi ( norm ‘ ( 𝐴 𝐵 ) ) = ( norm ‘ ( 𝐵 𝐴 ) )

Proof

Step Hyp Ref Expression
1 normsub.1 𝐴 ∈ ℋ
2 normsub.2 𝐵 ∈ ℋ
3 neg1cn - 1 ∈ ℂ
4 2 1 hvsubcli ( 𝐵 𝐴 ) ∈ ℋ
5 3 4 norm-iii-i ( norm ‘ ( - 1 · ( 𝐵 𝐴 ) ) ) = ( ( abs ‘ - 1 ) · ( norm ‘ ( 𝐵 𝐴 ) ) )
6 2 1 hvnegdii ( - 1 · ( 𝐵 𝐴 ) ) = ( 𝐴 𝐵 )
7 6 fveq2i ( norm ‘ ( - 1 · ( 𝐵 𝐴 ) ) ) = ( norm ‘ ( 𝐴 𝐵 ) )
8 ax-1cn 1 ∈ ℂ
9 8 absnegi ( abs ‘ - 1 ) = ( abs ‘ 1 )
10 abs1 ( abs ‘ 1 ) = 1
11 9 10 eqtri ( abs ‘ - 1 ) = 1
12 11 oveq1i ( ( abs ‘ - 1 ) · ( norm ‘ ( 𝐵 𝐴 ) ) ) = ( 1 · ( norm ‘ ( 𝐵 𝐴 ) ) )
13 4 normcli ( norm ‘ ( 𝐵 𝐴 ) ) ∈ ℝ
14 13 recni ( norm ‘ ( 𝐵 𝐴 ) ) ∈ ℂ
15 14 mulid2i ( 1 · ( norm ‘ ( 𝐵 𝐴 ) ) ) = ( norm ‘ ( 𝐵 𝐴 ) )
16 12 15 eqtri ( ( abs ‘ - 1 ) · ( norm ‘ ( 𝐵 𝐴 ) ) ) = ( norm ‘ ( 𝐵 𝐴 ) )
17 5 7 16 3eqtr3i ( norm ‘ ( 𝐴 𝐵 ) ) = ( norm ‘ ( 𝐵 𝐴 ) )