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Metamath Proof Explorer

Theorem xmetlecl

Description: Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015)

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

Proof

Step Hyp Ref Expression
1 xmetcl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
2 1 3expb 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A D B ) e. RR* )
3 2 3adant3 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR* )
4 simp3l 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> C e. RR )
5 xmetge0 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
6 5 3expb 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) ) -> 0 <_ ( A D B ) )
7 6 3adant3 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> 0 <_ ( A D B ) )
8 simp3r 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) <_ C )
9 xrrege0 
 |-  ( ( ( ( A D B ) e. RR* /\ C e. RR ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR )
10 3 4 7 8 9 syl22anc 
 |-  ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR )