Theorem distspace 12509

Description: A set X together with a (distance) function D which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set X equipped with a distance D, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)

Assertion

Ref	Expression
distspace	|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )

Proof of Theorem distspace

Step	Hyp	Ref	Expression
1		psmetf 12499	. . . 4 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
2	1	3ad2ant1 1002	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> D : ( X X. X ) --> RR* )
3		psmet0 12501	. . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
4	3	3adant3 1001	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
5	2, 4	jca 304	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) )
6		psmetge0 12505	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
7		psmetsym 12503	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
8	5, 6, 7	jca32 308	1 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   X. cxp 4537   -->wf 5119   ` cfv 5123  (class class class)co 5774   0cc0 7625   RR*cxr 7804   <_ cle 7806  PsMetcpsmet 12153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7716  ax-resscn 7717  ax-1cn 7718  ax-1re 7719  ax-icn 7720  ax-addcl 7721  ax-addrcl 7722  ax-mulcl 7723  ax-mulrcl 7724  ax-addcom 7725  ax-mulcom 7726  ax-addass 7727  ax-mulass 7728  ax-distr 7729  ax-i2m1 7730  ax-0lt1 7731  ax-1rid 7732  ax-0id 7733  ax-rnegex 7734  ax-precex 7735  ax-cnre 7736  ax-pre-ltirr 7737  ax-pre-lttrn 7739  ax-pre-apti 7740  ax-pre-ltadd 7741  ax-pre-mulgt0 7742

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-pnf 7807  df-mnf 7808  df-xr 7809  df-ltxr 7810  df-le 7811  df-sub 7940  df-neg 7941  df-2 8784  df-xadd 9565  df-psmet 12161

This theorem is referenced by: (None)