Theorem remetdval 14707

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 5921	. . 3 |- ( A D B ) = ( D ` <. A , B >. )
2		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
3	2	fveq1i 5555	. . 3 |- ( D ` <. A , B >. ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
4	1, 3	eqtri 2214	. 2 |- ( A D B ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
5		opelxpi 4691	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6		fvres 5578	. . . 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 5921	. . . 4 |- ( A ( abs o. - ) B ) = ( ( abs o. - ) ` <. A , B >. )
9		recn 8005	. . . . 5 |- ( A e. RR -> A e. CC )
10		recn 8005	. . . . 5 |- ( B e. RR -> B e. CC )
11		eqid 2193	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
12	11	cnmetdval 14697	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ( abs o. - ) B ) = ( abs ` ( A - B ) ) )
13	9, 10, 12	syl2an 289	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A ( abs o. - ) B ) = ( abs ` ( A - B ) ) )
14	8, 13	eqtr3id 2240	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs o. - ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
15	7, 14	eqtrd 2226	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
16	4, 15	eqtrid 2238	1 |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   <.cop 3621   X. cxp 4657   |` cres 4661   o. ccom 4663   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   - cmin 8190   abscabs 11141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-sub 8192

This theorem is referenced by:  bl2ioo  14710