Theorem remetdval 12899

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 5821	. . 3 |- ( A D B ) = ( D ` <. A , B >. )
2		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
3	2	fveq1i 5466	. . 3 |- ( D ` <. A , B >. ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
4	1, 3	eqtri 2178	. 2 |- ( A D B ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
5		opelxpi 4615	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6		fvres 5489	. . . 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 5821	. . . 4 |- ( A ( abs o. - ) B ) = ( ( abs o. - ) ` <. A , B >. )
9		recn 7848	. . . . 5 |- ( A e. RR -> A e. CC )
10		recn 7848	. . . . 5 |- ( B e. RR -> B e. CC )
11		eqid 2157	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
12	11	cnmetdval 12889	. . . . 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 2204	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs o. - ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
15	7, 14	eqtrd 2190	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
16	4, 15	syl5eq 2202	1 |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   <.cop 3563   X. cxp 4581   |` cres 4585   o. ccom 4587   ` cfv 5167  (class class class)co 5818   CCcc 7713   RRcr 7714   - cmin 8029   abscabs 10879

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-resscn 7807  ax-1cn 7808  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-addcom 7815  ax-addass 7817  ax-distr 7819  ax-i2m1 7820  ax-0id 7823  ax-rnegex 7824  ax-cnre 7826

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-sub 8031

This theorem is referenced by:  bl2ioo  12902