Theorem psmetxrge0 14284

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 14277	. . 3 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
2	1	ffnd 5385	. 2 |- ( D e. (PsMet ` X ) -> D Fn ( X X. X ) )
3	1	ffvelcdmda 5671	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. RR* )
4		elxp6 6193	. . . . . . . 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 14283	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) -> 0 <_ ( ( 1st ` a ) D ( 2nd ` a ) ) )
7	6	3expb 1206	. . . . . . 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 6199	. . . . . . . . 9 |- ( a e. ( X X. X ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
10	9	fveq2d 5538	. . . . . . . 8 |- ( a e. ( X X. X ) -> ( D ` a ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
11		df-ov 5898	. . . . . . . 8 |- ( ( 1st ` a ) D ( 2nd ` a ) ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
12	10, 11	eqtr4di 2240	. . . . . . 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 4046	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> 0 <_ ( D ` a ) )
15		elxrge0 10007	. . . . 5 |- ( ( D ` a ) e. ( 0 [,] +oo ) <-> ( ( D ` a ) e. RR* /\ 0 <_ ( D ` a ) ) )
16	3, 14, 15	sylanbrc 417	. . . 4 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. ( 0 [,] +oo ) )
17	16	ralrimiva 2563	. . 3 |- ( D e. (PsMet ` X ) -> A. a e. ( X X. X ) ( D ` a ) e. ( 0 [,] +oo ) )
18		fnfvrnss 5696	. . 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 411	. 2 |- ( D e. (PsMet ` X ) -> ran D C_ ( 0 [,] +oo ) )
20		df-f 5239	. 2 |- ( D : ( X X. X ) --> ( 0 [,] +oo ) <-> ( D Fn ( X X. X ) /\ ran D C_ ( 0 [,] +oo ) ) )
21	2, 19, 20	sylanbrc 417	1 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   <.cop 3610   class class class wbr 4018   X. cxp 4642   ran crn 4645   Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5895   1stc1st 6162   2ndc2nd 6163   0cc0 7840   +oocpnf 8018   RR*cxr 8020   <_ cle 8022   [,]cicc 9920  PsMetcpsmet 13845

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956  ax-pre-mulgt0 7957

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-map 6675  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-2 9007  df-xadd 9802  df-icc 9924  df-psmet 13853

This theorem is referenced by: (None)