Theorem xmeteq0 13153

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 13137	. . . . . . 7 |- Rel *Met
2		relelfvdm 5528	. . . . . . 7 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
3	1, 2	mpan 422	. . . . . 6 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
4		isxmet 13139	. . . . . 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 2534	. . . 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 5860	. . . . . 6 |- ( x = A -> ( x D y ) = ( A D y ) )
11	10	eqeq1d 2179	. . . . 5 |- ( x = A -> ( ( x D y ) = 0 <-> ( A D y ) = 0 ) )
12		eqeq1 2177	. . . . 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 5861	. . . . . 6 |- ( y = B -> ( A D y ) = ( A D B ) )
15	14	eqeq1d 2179	. . . . 5 |- ( y = B -> ( ( A D y ) = 0 <-> ( A D B ) = 0 ) )
16		eqeq2 2180	. . . . 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 2847	. . 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 1196	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 973   = wceq 1348   e. wcel 2141   A.wral 2448   class class class wbr 3989   X. cxp 4609   dom cdm 4611   Rel wrel 4616   -->wf 5194   ` cfv 5198  (class class class)co 5853   0cc0 7774   RR*cxr 7953   <_ cle 7955   +ecxad 9727   *Metcxmet 12774

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-iun 3875  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-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-xmet 12782

This theorem is referenced by:  meteq0  13154  xmet0  13157  xmetres2  13173  xblss2  13199  xmseq0  13262  comet  13293  xmetxp  13301