Theorem psmet0 15351

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 14852	. . . . . . . . 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 5782	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. _V )
3		ispsmet 15347	. . . . . . . 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 2638	. . . 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 2623	. 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 6093	. . . 4 |- ( a = A -> ( a D a ) = ( A D A ) )
12	11	eqeq1d 2247	. . 3 |- ( a = A -> ( ( a D a ) = 0 <-> ( A D A ) = 0 ) )
13	12	rspcv 2925	. 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 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   class class class wbr 4125   X. cxp 4767   -->wf 5368   ` cfv 5372  (class class class)co 6075   ^m cmap 6912   0cc0 8169   RR*cxr 8349   <_ cle 8351   +ecxad 10151  PsMetcpsmet 14844

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-psmet 14852

This theorem is referenced by:  psmetsym  15353  psmetge0  15355  psmetres2  15357  distspace  15359  xblcntrps  15437  ssblps  15449