Theorem xmeteq0 12999

Description: The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmeteq0	|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )

Proof of Theorem xmeteq0

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetrel 12983	. . . . . . 7 |- Rel *Met
2		relelfvdm 5518	. . . . . . 7 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
3	1, 2	mpan 421	. . . . . 6 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
4		isxmet 12985	. . . . . 6 |- ( X e. dom *Met -> ( D e. ( *Met ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) ) )
5	3, 4	syl 14	. . . . 5 |- ( D e. ( *Met ` X ) -> ( D e. ( *Met ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) ) )
6	5	ibi 175	. . . 4 |- ( D e. ( *Met ` X ) -> ( D : ( X X. X ) --> RR* /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) )
7		simpl 108	. . . . 5 |- ( ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) -> ( ( x D y ) = 0 <-> x = y ) )
8	7	2ralimi 2530	. . . 4 |- ( A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) -> A. x e. X A. y e. X ( ( x D y ) = 0 <-> x = y ) )
9	6, 8	simpl2im 384	. . 3 |- ( D e. ( *Met ` X ) -> A. x e. X A. y e. X ( ( x D y ) = 0 <-> x = y ) )
10		oveq1 5849	. . . . . 6 |- ( x = A -> ( x D y ) = ( A D y ) )
11	10	eqeq1d 2174	. . . . 5 |- ( x = A -> ( ( x D y ) = 0 <-> ( A D y ) = 0 ) )
12		eqeq1 2172	. . . . 5 |- ( x = A -> ( x = y <-> A = y ) )
13	11, 12	bibi12d 234	. . . 4 |- ( x = A -> ( ( ( x D y ) = 0 <-> x = y ) <-> ( ( A D y ) = 0 <-> A = y ) ) )
14		oveq2 5850	. . . . . 6 |- ( y = B -> ( A D y ) = ( A D B ) )
15	14	eqeq1d 2174	. . . . 5 |- ( y = B -> ( ( A D y ) = 0 <-> ( A D B ) = 0 ) )
16		eqeq2 2175	. . . . 5 |- ( y = B -> ( A = y <-> A = B ) )
17	15, 16	bibi12d 234	. . . 4 |- ( y = B -> ( ( ( A D y ) = 0 <-> A = y ) <-> ( ( A D B ) = 0 <-> A = B ) ) )
18	13, 17	rspc2v 2843	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( ( x D y ) = 0 <-> x = y ) -> ( ( A D B ) = 0 <-> A = B ) ) )
19	9, 18	syl5com 29	. 2 |- ( D e. ( *Met ` X ) -> ( ( A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) ) )
20	19	3impib 1191	1 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982   X. cxp 4602   dom cdm 4604   Rel wrel 4609   -->wf 5184   ` cfv 5188  (class class class)co 5842   0cc0 7753   RR*cxr 7932   <_ cle 7934   +ecxad 9706   *Metcxmet 12620

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-xmet 12628

This theorem is referenced by:  meteq0  13000  xmet0  13003  xmetres2  13019  xblss2  13045  xmseq0  13108  comet  13139  xmetxp  13147