Theorem xmeteq0 14679

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 14663	. . . . . . 7 |- Rel *Met
2		relelfvdm 5593	. . . . . . 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 14665	. . . . . 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 2561	. . . 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 5932	. . . . . 6 |- ( x = A -> ( x D y ) = ( A D y ) )
11	10	eqeq1d 2205	. . . . 5 |- ( x = A -> ( ( x D y ) = 0 <-> ( A D y ) = 0 ) )
12		eqeq1 2203	. . . . 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 5933	. . . . . 6 |- ( y = B -> ( A D y ) = ( A D B ) )
15	14	eqeq1d 2205	. . . . 5 |- ( y = B -> ( ( A D y ) = 0 <-> ( A D B ) = 0 ) )
16		eqeq2 2206	. . . . 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 2881	. . 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 1203	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 980   = wceq 1364   e. wcel 2167   A.wral 2475   class class class wbr 4034   X. cxp 4662   dom cdm 4664   Rel wrel 4669   -->wf 5255   ` cfv 5259  (class class class)co 5925   0cc0 7896   RR*cxr 8077   <_ cle 8079   +ecxad 9862   *Metcxmet 14168

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xmet 14176

This theorem is referenced by:  meteq0  14680  xmet0  14683  xmetres2  14699  xblss2  14725  xmseq0  14788  comet  14819  xmetxp  14827