Theorem xmeteq0 14538

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 14522	. . . . . . 7 |- Rel *Met
2		relelfvdm 5587	. . . . . . 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 14524	. . . . . 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 2558	. . . 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 5926	. . . . . 6 |- ( x = A -> ( x D y ) = ( A D y ) )
11	10	eqeq1d 2202	. . . . 5 |- ( x = A -> ( ( x D y ) = 0 <-> ( A D y ) = 0 ) )
12		eqeq1 2200	. . . . 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 5927	. . . . . 6 |- ( y = B -> ( A D y ) = ( A D B ) )
15	14	eqeq1d 2202	. . . . 5 |- ( y = B -> ( ( A D y ) = 0 <-> ( A D B ) = 0 ) )
16		eqeq2 2203	. . . . 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 2878	. . 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 2164   A.wral 2472   class class class wbr 4030   X. cxp 4658   dom cdm 4660   Rel wrel 4665   -->wf 5251   ` cfv 5255  (class class class)co 5919   0cc0 7874   RR*cxr 8055   <_ cle 8057   +ecxad 9839   *Metcxmet 14035

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-xr 8060  df-xmet 14043

This theorem is referenced by:  meteq0  14539  xmet0  14542  xmetres2  14558  xblss2  14584  xmseq0  14647  comet  14678  xmetxp  14686