Theorem xmeteq0 15170

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 15154	. . . . . . 7 |- Rel *Met
2		relelfvdm 5680	. . . . . . 7 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
3	1, 2	mpan 424	. . . . . 6 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
4		isxmet 15156	. . . . . 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 176	. . . 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 109	. . . . 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 2597	. . . 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 386	. . 3 |- ( D e. ( *Met ` X ) -> A. x e. X A. y e. X ( ( x D y ) = 0 <-> x = y ) )
10		oveq1 6035	. . . . . 6 |- ( x = A -> ( x D y ) = ( A D y ) )
11	10	eqeq1d 2240	. . . . 5 |- ( x = A -> ( ( x D y ) = 0 <-> ( A D y ) = 0 ) )
12		eqeq1 2238	. . . . 5 |- ( x = A -> ( x = y <-> A = y ) )
13	11, 12	bibi12d 235	. . . 4 |- ( x = A -> ( ( ( x D y ) = 0 <-> x = y ) <-> ( ( A D y ) = 0 <-> A = y ) ) )
14		oveq2 6036	. . . . . 6 |- ( y = B -> ( A D y ) = ( A D B ) )
15	14	eqeq1d 2240	. . . . 5 |- ( y = B -> ( ( A D y ) = 0 <-> ( A D B ) = 0 ) )
16		eqeq2 2241	. . . . 5 |- ( y = B -> ( A = y <-> A = B ) )
17	15, 16	bibi12d 235	. . . 4 |- ( y = B -> ( ( ( A D y ) = 0 <-> A = y ) <-> ( ( A D B ) = 0 <-> A = B ) ) )
18	13, 17	rspc2v 2924	. . 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 1228	1 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   class class class wbr 4093   X. cxp 4729   dom cdm 4731   Rel wrel 4736   -->wf 5329   ` cfv 5333  (class class class)co 6028   0cc0 8092   RR*cxr 8272   <_ cle 8274   +ecxad 10066   *Metcxmet 14632

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-xmet 14640

This theorem is referenced by:  meteq0  15171  xmet0  15174  xmetres2  15190  xblss2  15216  xmseq0  15279  comet  15310  xmetxp  15318