Theorem psmetxrge0 14652

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 14645	. . 3 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
2	1	ffnd 5411	. 2 |- ( D e. (PsMet ` X ) -> D Fn ( X X. X ) )
3	1	ffvelcdmda 5700	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. RR* )
4		elxp6 6236	. . . . . . . 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 14651	. . . . . . . 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 6242	. . . . . . . . 9 |- ( a e. ( X X. X ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
10	9	fveq2d 5565	. . . . . . . 8 |- ( a e. ( X X. X ) -> ( D ` a ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
11		df-ov 5928	. . . . . . . 8 |- ( ( 1st ` a ) D ( 2nd ` a ) ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
12	10, 11	eqtr4di 2247	. . . . . . 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 4062	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> 0 <_ ( D ` a ) )
15		elxrge0 10070	. . . . 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 2570	. . 3 |- ( D e. (PsMet ` X ) -> A. a e. ( X X. X ) ( D ` a ) e. ( 0 [,] +oo ) )
18		fnfvrnss 5725	. . 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 5263	. 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 2167   A.wral 2475   C_ wss 3157   <.cop 3626   class class class wbr 4034   X. cxp 4662   ran crn 4665   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   0cc0 7896   +oocpnf 8075   RR*cxr 8077   <_ cle 8079   [,]cicc 9983  PsMetcpsmet 14167

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-2 9066  df-xadd 9865  df-icc 9987  df-psmet 14175

This theorem is referenced by: (None)