Theorem meteq0 7751

Description: The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4.

Hypothesis

Ref	Expression
metf.1	|- X = dom dom D

Assertion

Ref	Expression
meteq0	|- ((D e. Met /\ A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B))

Proof of Theorem meteq0

Step	Hyp	Ref	Expression
1		metf.1	. . . . 5 |- X = dom dom D
2	1	metflem 7745	. . . 4 |- (D e. Met -> (D:(X X. X)-->RR /\ A.x e. X A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy)))))
3	2	pm3.27d 325	. . 3 |- (D e. Met -> A.x e. X A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))))
4		pm3.26 319	. . . . 5 |- ((((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))) -> ((xDy) = 0 <-> x = y))
5	4	r19.20si 1698	. . . 4 |- (A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))) -> A.y e. X ((xDy) = 0 <-> x = y))
6	5	r19.20si 1698	. . 3 |- (A.x e. X A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))) -> A.x e. X A.y e. X ((xDy) = 0 <-> x = y))
7		opreq1 3953	. . . . . . 7 |- (x = A -> (xDy) = (ADy))
8	7	eqeq1d 1475	. . . . . 6 |- (x = A -> ((xDy) = 0 <-> (ADy) = 0))
9		eqeq1 1473	. . . . . 6 |- (x = A -> (x = y <-> A = y))
10	8, 9	bibi12d 627	. . . . 5 |- (x = A -> (((xDy) = 0 <-> x = y) <-> ((ADy) = 0 <-> A = y)))
11		opreq2 3954	. . . . . . 7 |- (y = B -> (ADy) = (ADB))
12	11	eqeq1d 1475	. . . . . 6 |- (y = B -> ((ADy) = 0 <-> (ADB) = 0))
13		eqeq2 1476	. . . . . 6 |- (y = B -> (A = y <-> A = B))
14	12, 13	bibi12d 627	. . . . 5 |- (y = B -> (((ADy) = 0 <-> A = y) <-> ((ADB) = 0 <-> A = B)))
15	10, 14	rcla42v 1871	. . . 4 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X ((xDy) = 0 <-> x = y) -> ((ADB) = 0 <-> A = B)))
16	15	com12 11	. . 3 |- (A.x e. X A.y e. X ((xDy) = 0 <-> x = y) -> ((A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B)))
17	3, 6, 16	3syl 20	. 2 |- (D e. Met -> ((A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B)))
18	17	3impib 829	1 |- ((D e. Met /\ A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637   class class class wbr 2609   X. cxp 3158  dom cdm 3160  -->wf 3168  (class class class)co 3948  RRcr 5205  0cc0 5206   + caddc 5209   <_ cle 5267  Metcme 7728

This theorem is referenced by:  met0 7754  metgt0 7761  metxp 7774  lmuni 7886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-met 7732

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