Metamath Proof Explorer

Theorem abvtri

Description: An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014)

Ref Expression
Hypotheses abvf.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
abvf.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
abvtri.p ⊢ + = ( +g ‘ 𝑅 )
Assertion abvtri ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 abvf.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
2 abvf.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 abvtri.p ⊢ + = ( +g ‘ 𝑅 )
4 1 abvrcl ⊢ ( 𝐹 ∈ 𝐴 → 𝑅 ∈ Ring )
5 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
6 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
7 1 2 3 5 6 isabv ⊢ ( 𝑅 ∈ Ring → ( 𝐹 ∈ 𝐴 ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = ( 0g ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) )
8 4 7 syl ⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∈ 𝐴 ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = ( 0g ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) )
9 8 ibi ⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = ( 0g ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) ) )
10 simpr ⊢ ( ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) )
11 10 ralimi ⊢ ( ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) )
12 11 adantl ⊢ ( ( ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = ( 0g ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) → ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) )
13 12 ralimi ⊢ ( ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = ( 0g ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) )
14 9 13 simpl2im ⊢ ( 𝐹 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) )
15 fvoveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) )
16 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) )
17 16 oveq1d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑦 ) ) )
18 15 17 breq12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑦 ) ) ) )
19 oveq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑋 + 𝑦 ) = ( 𝑋 + 𝑌 ) )
20 19 fveq2d ⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) )
21 fveq2 ⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑌 ) )
22 21 oveq2d ⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )
23 20 22 breq12d ⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) )
24 18 23 rspc2v ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) )
25 14 24 syl5com ⊢ ( 𝐹 ∈ 𝐴 → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) )
26 25 3impib ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )