Theorem psmetxrge0 15323

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 15316	. . 3 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
2	1	ffnd 5514	. 2 |- ( D e. (PsMet ` X ) -> D Fn ( X X. X ) )
3	1	ffvelcdmda 5817	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> ( D ` a ) e. RR* )
4		elxp6 6376	. . . . . . . 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 15322	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( 1st ` a ) e. X /\ ( 2nd ` a ) e. X ) -> 0 <_ ( ( 1st ` a ) D ( 2nd ` a ) ) )
7	6	3expb 1231	. . . . . . 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 6382	. . . . . . . . 9 |- ( a e. ( X X. X ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
10	9	fveq2d 5679	. . . . . . . 8 |- ( a e. ( X X. X ) -> ( D ` a ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
11		df-ov 6061	. . . . . . . 8 |- ( ( 1st ` a ) D ( 2nd ` a ) ) = ( D ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
12	10, 11	eqtr4di 2285	. . . . . . 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 4142	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ a e. ( X X. X ) ) -> 0 <_ ( D ` a ) )
15		elxrge0 10330	. . . . 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 2617	. . 3 |- ( D e. (PsMet ` X ) -> A. a e. ( X X. X ) ( D ` a ) e. ( 0 [,] +oo ) )
18		fnfvrnss 5842	. . 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 5361	. 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 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   <.cop 3697   class class class wbr 4114   X. cxp 4752   ran crn 4755   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   0cc0 8143   +oocpnf 8321   RR*cxr 8323   <_ cle 8325   [,]cicc 10243  PsMetcpsmet 14809

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-2 9313  df-xadd 10125  df-icc 10247  df-psmet 14817

This theorem is referenced by: (None)