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Metamath Proof Explorer

Theorem abvfge0

Description: An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014)

Ref Expression
Hypotheses abvf.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
abvf.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion abvfge0 ( 𝐹 ∈ 𝐴 → 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) )

Proof

Step Hyp Ref Expression
1 abvf.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
2 abvf.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 1 abvrcl ⊢ ( 𝐹 ∈ 𝐴 → 𝑅 ∈ Ring )
4 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
5 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
6 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
7 1 2 4 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 ( +g ‘ 𝑅 ) 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) )
8 3 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 ( +g ‘ 𝑅 ) 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 ( +g ‘ 𝑅 ) 𝑦 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) ) )
10 9 simpld ⊢ ( 𝐹 ∈ 𝐴 → 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) )