Theorem remetdval 15221

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 6004	. . 3 |- ( A D B ) = ( D ` <. A , B >. )
2		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
3	2	fveq1i 5628	. . 3 |- ( D ` <. A , B >. ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
4	1, 3	eqtri 2250	. 2 |- ( A D B ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
5		opelxpi 4751	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6		fvres 5651	. . . 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 6004	. . . 4 |- ( A ( abs o. - ) B ) = ( ( abs o. - ) ` <. A , B >. )
9		recn 8132	. . . . 5 |- ( A e. RR -> A e. CC )
10		recn 8132	. . . . 5 |- ( B e. RR -> B e. CC )
11		eqid 2229	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
12	11	cnmetdval 15203	. . . . 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 2276	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs o. - ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
15	7, 14	eqtrd 2262	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
16	4, 15	eqtrid 2274	1 |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   <.cop 3669   X. cxp 4717   |` cres 4721   o. ccom 4723   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   - cmin 8317   abscabs 11508

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-sub 8319

This theorem is referenced by:  bl2ioo  15224