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Metamath Proof Explorer

Theorem abvsubtri

Description: An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses abv0.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
abvneg.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
abvsubtri.p ⊢ − = ( -g ‘ 𝑅 )
Assertion abvsubtri ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 − 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 abv0.a ⊢ 𝐴 = ( AbsVal ‘ 𝑅 )
2 abvneg.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 abvsubtri.p ⊢ − = ( -g ‘ 𝑅 )
4 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
5 eqid ⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
6 2 4 5 3 grpsubval ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) )
7 6 3adant1 ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) )
8 7 fveq2d ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 − 𝑌 ) ) = ( 𝐹 ‘ ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) )
9 1 abvrcl ⊢ ( 𝐹 ∈ 𝐴 → 𝑅 ∈ Ring )
10 9 3ad2ant1 ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑅 ∈ Ring )
11 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
12 10 11 syl ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑅 ∈ Grp )
13 simp3 ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
14 2 5 grpinvcl ⊢ ( ( 𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 )
15 12 13 14 syl2anc ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 )
16 1 2 4 abvtri ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) )
17 15 16 syld3an3 ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) )
18 1 2 5 abvneg ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) = ( 𝐹 ‘ 𝑌 ) )
19 18 3adant2 ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) = ( 𝐹 ‘ 𝑌 ) )
20 19 oveq2d ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) = ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )
21 17 20 breqtrd ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ 𝑌 ) ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )
22 8 21 eqbrtrd ⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 − 𝑌 ) ) ≤ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) )