Theorem psmet0 13912

Description: The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
psmet0	|- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )

Proof of Theorem psmet0

Dummy variables a b c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psmet 13532	. . . . . . . . 9 |- PsMet = ( d e. _V |-> { e e. ( RR* ^m ( d X. d ) ) | A. a e. d ( ( a e a ) = 0 /\ A. b e. d A. c e. d ( a e b ) <_ ( ( c e a ) +e ( c e b ) ) ) } )
2	1	mptrcl 5600	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. _V )
3		ispsmet 13908	. . . . . . . 8 |- ( X e. _V -> ( D e. (PsMet ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. a e. X ( ( a D a ) = 0 /\ A. b e. X A. c e. X ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) ) ) ) )
4	2, 3	syl 14	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( D e. (PsMet ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. a e. X ( ( a D a ) = 0 /\ A. b e. X A. c e. X ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) ) ) ) )
5	4	ibi 176	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( D : ( X X. X ) --> RR* /\ A. a e. X ( ( a D a ) = 0 /\ A. b e. X A. c e. X ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) ) ) )
6	5	simprd 114	. . . . 5 |- ( D e. (PsMet ` X ) -> A. a e. X ( ( a D a ) = 0 /\ A. b e. X A. c e. X ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) ) )
7	6	r19.21bi 2565	. . . 4 |- ( ( D e. (PsMet ` X ) /\ a e. X ) -> ( ( a D a ) = 0 /\ A. b e. X A. c e. X ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) ) )
8	7	simpld 112	. . 3 |- ( ( D e. (PsMet ` X ) /\ a e. X ) -> ( a D a ) = 0 )
9	8	ralrimiva 2550	. 2 |- ( D e. (PsMet ` X ) -> A. a e. X ( a D a ) = 0 )
10		id 19	. . . . 5 |- ( a = A -> a = A )
11	10, 10	oveq12d 5895	. . . 4 |- ( a = A -> ( a D a ) = ( A D A ) )
12	11	eqeq1d 2186	. . 3 |- ( a = A -> ( ( a D a ) = 0 <-> ( A D A ) = 0 ) )
13	12	rspcv 2839	. 2 |- ( A e. X -> ( A. a e. X ( a D a ) = 0 -> ( A D A ) = 0 ) )
14	9, 13	mpan9 281	1 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   {crab 2459   _Vcvv 2739   class class class wbr 4005   X. cxp 4626   -->wf 5214   ` cfv 5218  (class class class)co 5877   ^m cmap 6650   0cc0 7813   RR*cxr 7993   <_ cle 7995   +ecxad 9772  PsMetcpsmet 13524

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-map 6652  df-pnf 7996  df-mnf 7997  df-xr 7998  df-psmet 13532

This theorem is referenced by:  psmetsym  13914  psmetge0  13916  psmetres2  13918  distspace  13920  xblcntrps  13998  ssblps  14010