Theorem psmet0 15192

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 14691	. . . . . . . . 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 5760	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. _V )
3		ispsmet 15188	. . . . . . . 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 2630	. . . 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 2615	. 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 6068	. . . 4 |- ( a = A -> ( a D a ) = ( A D A ) )
12	11	eqeq1d 2241	. . 3 |- ( a = A -> ( ( a D a ) = 0 <-> ( A D A ) = 0 ) )
13	12	rspcv 2917	. 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 1398   e. wcel 2203   A.wral 2520   {crab 2524   _Vcvv 2813   class class class wbr 4109   X. cxp 4747   -->wf 5348   ` cfv 5352  (class class class)co 6050   ^m cmap 6882   0cc0 8127   RR*cxr 8307   <_ cle 8309   +ecxad 10103  PsMetcpsmet 14683

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-psmet 14691

This theorem is referenced by:  psmetsym  15194  psmetge0  15196  psmetres2  15198  distspace  15200  xblcntrps  15278  ssblps  15290