Theorem remetdval 15019

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 5947	. . 3 |- ( A D B ) = ( D ` <. A , B >. )
2		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
3	2	fveq1i 5577	. . 3 |- ( D ` <. A , B >. ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
4	1, 3	eqtri 2226	. 2 |- ( A D B ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. )
5		opelxpi 4707	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6		fvres 5600	. . . 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 5947	. . . 4 |- ( A ( abs o. - ) B ) = ( ( abs o. - ) ` <. A , B >. )
9		recn 8058	. . . . 5 |- ( A e. RR -> A e. CC )
10		recn 8058	. . . . 5 |- ( B e. RR -> B e. CC )
11		eqid 2205	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
12	11	cnmetdval 15001	. . . . 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 2252	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs o. - ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
15	7, 14	eqtrd 2238	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) ` <. A , B >. ) = ( abs ` ( A - B ) ) )
16	4, 15	eqtrid 2250	1 |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   <.cop 3636   X. cxp 4673   |` cres 4677   o. ccom 4679   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   - cmin 8243   abscabs 11308

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-resscn 8017  ax-1cn 8018  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-sub 8245

This theorem is referenced by:  bl2ioo  15022