Theorem psmetxrge0 15356

Description: The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)

Assertion

Ref	Expression
psmetxrge0	|- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> ( 0 [,] +oo ) )

Proof of Theorem psmetxrge0

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psmetf 15349	. . 3 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
2	1	ffnd 5529	. 2 |- ( D e. (PsMet ` X ) -> D Fn ( X X. X ) )
3	1	ffvelcdmda 5834	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. RR* )
4		elxp6 6393	. . . . . . . 8 |- ( a e. ( X X. X ) <-> ( a = <. ( 1st ` a ) , ( 2nd ` a ) >. /\ ( ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) ) )
5	4	simprbi 275	. . . . . . 7 |- ( a e. ( X X. X ) -> ( ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) )
6		psmetge0 15355	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) -> 0 <_ ( ( 1st ` a ) D ( 2nd ` a ) ) )
7	6	3expb 1235	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) ) -> 0 <_ ( ( 1st ` a ) D ( 2nd ` a ) ) )
8	5, 7	sylan2 286	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> 0 <_ ( ( 1st ` a ) D ( 2nd ` a ) ) )
9		1st2nd2 6399	. . . . . . . . 9 |- ( a e. ( X X. X ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
10	9	fveq2d 5694	. . . . . . . 8 |- ( a e. ( X X. X ) -> ( D ` a ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
11		df-ov 6078	. . . . . . . 8 |- ( ( 1st ` a ) D ( 2nd ` a ) ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
12	10, 11	eqtr4di 2289	. . . . . . 7 |- ( a e. ( X X. X ) -> ( D ` a ) = ( ( 1st ` a ) D ( 2nd ` a ) ) )
13	12	adantl 277	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) = ( ( 1st ` a ) D ( 2nd ` a ) ) )
14	8, 13	breqtrrd 4153	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> 0 <_ ( D ` a ) )
15		elxrge0 10359	. . . . 5 |- ( ( D ` a ) e. ( 0 [,] +oo ) <-> ( ( D ` a ) e. RR* /\ 0 <_ ( D ` a ) ) )
16	3, 14, 15	sylanbrc 421	. . . 4 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. ( 0 [,] +oo ) )
17	16	ralrimiva 2623	. . 3 |- ( D e. (PsMet ` X ) -> A. a e. ( X X. X ) ( D ` a ) e. ( 0 [,] +oo ) )
18		fnfvrnss 5859	. . 3 |- ( ( D Fn ( X X. X ) /\ A. a e. ( X X. X ) ( D ` a ) e. ( 0 [,] +oo ) ) -> ran D C_ ( 0 [,] +oo ) )
19	2, 17, 18	syl2anc 415	. 2 |- ( D e. (PsMet ` X ) -> ran D C_ ( 0 [,] +oo ) )
20		df-f 5376	. 2 |- ( D : ( X X. X ) --> ( 0 [,] +oo ) <-> ( D Fn ( X X. X ) /\ ran D C_ ( 0 [,] +oo ) ) )
21	2, 19, 20	sylanbrc 421	1 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   <.cop 3708   class class class wbr 4125   X. cxp 4767   ran crn 4770   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   0cc0 8169   +oocpnf 8347   RR*cxr 8349   <_ cle 8351   [,]cicc 10272  PsMetcpsmet 14844

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-2 9342  df-xadd 10154  df-icc 10276  df-psmet 14852

This theorem is referenced by: (None)