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Metamath Proof Explorer

Theorem abs2dif

Description: Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007)

Ref Expression
Assertion abs2dif 
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )

Proof

Step Hyp Ref Expression
1 subid1 
 |-  ( A e. CC -> ( A - 0 ) = A )
2 1 fveq2d 
 |-  ( A e. CC -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
3 subid1 
 |-  ( B e. CC -> ( B - 0 ) = B )
4 3 fveq2d 
 |-  ( B e. CC -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
5 2 4 oveqan12d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) = ( ( abs ` A ) - ( abs ` B ) ) )
6 0cn 
 |-  0 e. CC
7 abs3dif 
 |-  ( ( A e. CC /\ 0 e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
8 6 7 mp3an2 
 |-  ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
9 subcl 
 |-  ( ( A e. CC /\ 0 e. CC ) -> ( A - 0 ) e. CC )
10 6 9 mpan2 
 |-  ( A e. CC -> ( A - 0 ) e. CC )
11 abscl 
 |-  ( ( A - 0 ) e. CC -> ( abs ` ( A - 0 ) ) e. RR )
12 10 11 syl 
 |-  ( A e. CC -> ( abs ` ( A - 0 ) ) e. RR )
13 subcl 
 |-  ( ( B e. CC /\ 0 e. CC ) -> ( B - 0 ) e. CC )
14 6 13 mpan2 
 |-  ( B e. CC -> ( B - 0 ) e. CC )
15 abscl 
 |-  ( ( B - 0 ) e. CC -> ( abs ` ( B - 0 ) ) e. RR )
16 14 15 syl 
 |-  ( B e. CC -> ( abs ` ( B - 0 ) ) e. RR )
17 12 16 anim12i 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR ) )
18 subcl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
19 abscl 
 |-  ( ( A - B ) e. CC -> ( abs ` ( A - B ) ) e. RR )
20 18 19 syl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) e. RR )
21 df-3an 
 |-  ( ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) <-> ( ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR ) /\ ( abs ` ( A - B ) ) e. RR ) )
22 17 20 21 sylanbrc 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) )
23 lesubadd 
 |-  ( ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) -> ( ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) <_ ( abs ` ( A - B ) ) <-> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) ) )
24 22 23 syl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) <_ ( abs ` ( A - B ) ) <-> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) ) )
25 8 24 mpbird 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) <_ ( abs ` ( A - B ) ) )
26 5 25 eqbrtrrd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )