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Metamath Proof Explorer

Theorem xmetge0

Description: The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmetge0 
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> D e. ( *Met ` X ) )
2 simp2 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> A e. X )
3 simp3 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> B e. X )
4 xmettri2 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X /\ B e. X ) ) -> ( B D B ) <_ ( ( A D B ) +e ( A D B ) ) )
5 1 2 3 3 4 syl13anc 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) <_ ( ( A D B ) +e ( A D B ) ) )
6 2re 
 |-  2 e. RR
7 rexr 
 |-  ( 2 e. RR -> 2 e. RR* )
8 xmul01 
 |-  ( 2 e. RR* -> ( 2 *e 0 ) = 0 )
9 6 7 8 mp2b 
 |-  ( 2 *e 0 ) = 0
10 xmet0 
 |-  ( ( D e. ( *Met ` X ) /\ B e. X ) -> ( B D B ) = 0 )
11 10 3adant2 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
12 9 11 eqtr4id 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( 2 *e 0 ) = ( B D B ) )
13 xmetcl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
14 x2times 
 |-  ( ( A D B ) e. RR* -> ( 2 *e ( A D B ) ) = ( ( A D B ) +e ( A D B ) ) )
15 13 14 syl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( 2 *e ( A D B ) ) = ( ( A D B ) +e ( A D B ) ) )
16 5 12 15 3brtr4d 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( 2 *e 0 ) <_ ( 2 *e ( A D B ) ) )
17 0xr 
 |-  0 e. RR*
18 2rp 
 |-  2 e. RR+
19 18 a1i 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 2 e. RR+ )
20 xlemul2 
 |-  ( ( 0 e. RR* /\ ( A D B ) e. RR* /\ 2 e. RR+ ) -> ( 0 <_ ( A D B ) <-> ( 2 *e 0 ) <_ ( 2 *e ( A D B ) ) ) )
21 17 13 19 20 mp3an2i 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( 0 <_ ( A D B ) <-> ( 2 *e 0 ) <_ ( 2 *e ( A D B ) ) ) )
22 16 21 mpbird 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )