Theorem remetdval 12708

Description: Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)

Hypothesis

Ref	Expression
remet.1	|- D = ( ( abs o. - ) |` ( RR X. RR ) )

Assertion

Ref	Expression
remetdval	|- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Proof of Theorem remetdval

Step	Hyp	Ref	Expression
1		df-ov 5777	. . 3 |- ( A D B ) = ( D ` <. A , B >. )
2		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
3	2	fveq1i 5422	. . 3 |- ( D ` <. A , B >. ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
4	1, 3	eqtri 2160	. 2 |- ( A D B ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
5		opelxpi 4571	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6		fvres 5445	. . . 4 |- ( <. A , B >. e. ( RR X. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( ( abs o. - ) ` <. A , B >. ) )
7	5, 6	syl 14	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( ( abs o. - ) ` <. A , B >. ) )
8		df-ov 5777	. . . 4 |- ( A ( abs o. - ) B ) = ( ( abs o. - ) ` <. A , B >. )
9		recn 7753	. . . . 5 |- ( A e. RR -> A e. CC )
10		recn 7753	. . . . 5 |- ( B e. RR -> B e. CC )
11		eqid 2139	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
12	11	cnmetdval 12698	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ( abs o. - ) B ) = ( abs ` ( A - B ) ) )
13	9, 10, 12	syl2an 287	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A ( abs o. - ) B ) = ( abs ` ( A - B ) ) )
14	8, 13	syl5eqr 2186	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs o. - ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
15	7, 14	eqtrd 2172	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
16	4, 15	syl5eq 2184	1 |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   <.cop 3530   X. cxp 4537   |` cres 4541   o. ccom 4543   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   - cmin 7933   abscabs 10769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-sub 7935

This theorem is referenced by:  bl2ioo  12711