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Metamath Proof Explorer

Theorem norm-ii

Description: Triangle inequality for norms. Theorem 3.3(ii) of Beran p. 97. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)

Ref Expression
Assertion norm-ii ⊢ A ∈ ℋ ∧ B ∈ ℋ → norm ℎ ⁡ A + ℎ B ≤ norm ℎ ⁡ A + norm ℎ ⁡ B

Proof

Step Hyp Ref Expression
1 fvoveq1 ⊢ A = if A ∈ ℋ A 0 ℎ → norm ℎ ⁡ A + ℎ B = norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ B
2 fveq2 ⊢ A = if A ∈ ℋ A 0 ℎ → norm ℎ ⁡ A = norm ℎ ⁡ if A ∈ ℋ A 0 ℎ
3 2 oveq1d ⊢ A = if A ∈ ℋ A 0 ℎ → norm ℎ ⁡ A + norm ℎ ⁡ B = norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ B
4 1 3 breq12d ⊢ A = if A ∈ ℋ A 0 ℎ → norm ℎ ⁡ A + ℎ B ≤ norm ℎ ⁡ A + norm ℎ ⁡ B ↔ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ B ≤ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ B
5 oveq2 ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℋ A 0 ℎ + ℎ B = if A ∈ ℋ A 0 ℎ + ℎ if B ∈ ℋ B 0 ℎ
6 5 fveq2d ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ B = norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ if B ∈ ℋ B 0 ℎ
7 fveq2 ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ B = norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
8 7 oveq2d ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ B = norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
9 6 8 breq12d ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ B ≤ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ B ↔ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ if B ∈ ℋ B 0 ℎ ≤ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
10 ifhvhv0 ⊢ if A ∈ ℋ A 0 ℎ ∈ ℋ
11 ifhvhv0 ⊢ if B ∈ ℋ B 0 ℎ ∈ ℋ
12 10 11 norm-ii-i ⊢ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + ℎ if B ∈ ℋ B 0 ℎ ≤ norm ℎ ⁡ if A ∈ ℋ A 0 ℎ + norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
13 4 9 12 dedth2h ⊢ A ∈ ℋ ∧ B ∈ ℋ → norm ℎ ⁡ A + ℎ B ≤ norm ℎ ⁡ A + norm ℎ ⁡ B