Theorem psmet0 22113

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. _V )
2		ispsmet 22109	. . . . . . . 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 ) ) ) ) ) )
3	1, 2	syl 17	. . . . . . 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 ) ) ) ) ) )
4	3	ibi 256	. . . . . 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 ) ) ) ) )
5	4	simprd 479	. . . . 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 ) ) ) )
6	5	r19.21bi 2932	. . . 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 ) ) ) )
7	6	simpld 475	. . 3 |- ( ( D e. (PsMet ` X ) /\ a e. X ) -> ( a D a ) = 0 )
8	7	ralrimiva 2966	. 2 |- ( D e. (PsMet ` X ) -> A. a e. X ( a D a ) = 0 )
9		id 22	. . . . 5 |- ( a = A -> a = A )
10	9, 9	oveq12d 6668	. . . 4 |- ( a = A -> ( a D a ) = ( A D A ) )
11	10	eqeq1d 2624	. . 3 |- ( a = A -> ( ( a D a ) = 0 <-> ( A D A ) = 0 ) )
12	11	rspcv 3305	. 2 |- ( A e. X -> ( A. a e. X ( a D a ) = 0 -> ( A D A ) = 0 ) )
13	8, 12	mpan9 486	1 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-psmet 19738

This theorem is referenced by:  psmetsym  22115  psmetge0  22117  psmetres2  22119  distspace  22121  xblcntrps  22215  ssblps  22227  metustid  22359  metider  29937  pstmfval  29939