Metamath Proof Explorer


Theorem isabv

Description: Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014)

Ref Expression
Hypotheses abvfval.a 𝐴 = ( AbsVal ‘ 𝑅 )
abvfval.b 𝐵 = ( Base ‘ 𝑅 )
abvfval.p + = ( +g𝑅 )
abvfval.t · = ( .r𝑅 )
abvfval.z 0 = ( 0g𝑅 )
Assertion isabv ( 𝑅 ∈ Ring → ( 𝐹𝐴 ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 abvfval.a 𝐴 = ( AbsVal ‘ 𝑅 )
2 abvfval.b 𝐵 = ( Base ‘ 𝑅 )
3 abvfval.p + = ( +g𝑅 )
4 abvfval.t · = ( .r𝑅 )
5 abvfval.z 0 = ( 0g𝑅 )
6 1 2 3 4 5 abvfval ( 𝑅 ∈ Ring → 𝐴 = { 𝑓 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∣ ∀ 𝑥𝐵 ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) } )
7 6 eleq2d ( 𝑅 ∈ Ring → ( 𝐹𝐴𝐹 ∈ { 𝑓 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∣ ∀ 𝑥𝐵 ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) } ) )
8 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑥 ) = ( 𝐹𝑥 ) )
9 8 eqeq1d ( 𝑓 = 𝐹 → ( ( 𝑓𝑥 ) = 0 ↔ ( 𝐹𝑥 ) = 0 ) )
10 9 bibi1d ( 𝑓 = 𝐹 → ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ↔ ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ) )
11 fveq1 ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) )
12 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑦 ) = ( 𝐹𝑦 ) )
13 8 12 oveq12d ( 𝑓 = 𝐹 → ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) )
14 11 13 eqeq12d ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ) )
15 fveq1 ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) )
16 8 12 oveq12d ( 𝑓 = 𝐹 → ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) = ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) )
17 15 16 breq12d ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) )
18 14 17 anbi12d ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ↔ ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) )
19 18 ralbidv ( 𝑓 = 𝐹 → ( ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ↔ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) )
20 10 19 anbi12d ( 𝑓 = 𝐹 → ( ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) ↔ ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) )
21 20 ralbidv ( 𝑓 = 𝐹 → ( ∀ 𝑥𝐵 ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) ↔ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) )
22 21 elrab ( 𝐹 ∈ { 𝑓 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∣ ∀ 𝑥𝐵 ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) } ↔ ( 𝐹 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) )
23 ovex ( 0 [,) +∞ ) ∈ V
24 2 fvexi 𝐵 ∈ V
25 23 24 elmap ( 𝐹 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) )
26 25 anbi1i ( ( 𝐹 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) )
27 22 26 bitri ( 𝐹 ∈ { 𝑓 ∈ ( ( 0 [,) +∞ ) ↑m 𝐵 ) ∣ ∀ 𝑥𝐵 ( ( ( 𝑓𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓𝑥 ) · ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝑓𝑥 ) + ( 𝑓𝑦 ) ) ) ) } ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) )
28 7 27 syl6bb ( 𝑅 ∈ Ring → ( 𝐹𝐴 ↔ ( 𝐹 : 𝐵 ⟶ ( 0 [,) +∞ ) ∧ ∀ 𝑥𝐵 ( ( ( 𝐹𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹𝑥 ) · ( 𝐹𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ≤ ( ( 𝐹𝑥 ) + ( 𝐹𝑦 ) ) ) ) ) ) )