Theorem psmet0 13121

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 12781	. . . . . . . . 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 5578	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. _V )
3		ispsmet 13117	. . . . . . . 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 175	. . . . . 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 113	. . . . 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 2558	. . . 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 111	. . 3 |- ( ( D e. (PsMet ` X ) /\ a e. X ) -> ( a D a ) = 0 )
9	8	ralrimiva 2543	. 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 5871	. . . 4 |- ( a = A -> ( a D a ) = ( A D A ) )
12	11	eqeq1d 2179	. . 3 |- ( a = A -> ( ( a D a ) = 0 <-> ( A D A ) = 0 ) )
13	12	rspcv 2830	. 2 |- ( A e. X -> ( A. a e. X ( a D a ) = 0 -> ( A D A ) = 0 ) )
14	9, 13	mpan9 279	1 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730   class class class wbr 3989   X. cxp 4609   -->wf 5194   ` cfv 5198  (class class class)co 5853   ^m cmap 6626   0cc0 7774   RR*cxr 7953   <_ cle 7955   +ecxad 9727  PsMetcpsmet 12773

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-psmet 12781

This theorem is referenced by:  psmetsym  13123  psmetge0  13125  psmetres2  13127  distspace  13129  xblcntrps  13207  ssblps  13219