Theorem xmeter 15537

Description: The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Hypothesis

Ref	Expression
xmeter.1	|- .~ = ( `' D " RR )

Assertion

Ref	Expression
xmeter	|- ( D e. ( *Met ` X ) -> .~ Er X )

Proof of Theorem xmeter

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmeter.1	. . . . 5 |- .~ = ( `' D " RR )
2		cnvimass 5150	. . . . 5 |- ( `' D " RR ) C_ dom D
3	1, 2	eqsstri 3280	. . . 4 |- .~ C_ dom D
4		xmetf 15451	. . . 4 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
5	3, 4	fssdm 5549	. . 3 |- ( D e. ( *Met ` X ) -> .~ C_ ( X X. X ) )
6		relxp 4884	. . 3 |- Rel ( X X. X )
7		relss 4862	. . 3 |- ( .~ C_ ( X X. X ) -> ( Rel ( X X. X ) -> Rel .~ ) )
8	5, 6, 7	mpisyl 1496	. 2 |- ( D e. ( *Met ` X ) -> Rel .~ )
9	1	xmeterval 15536	. . . . 5 |- ( D e. ( *Met ` X ) -> ( x .~ y <-> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) ) )
10	9	biimpa 296	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) )
11	10	simp2d 1041	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y e. X )
12	10	simp1d 1040	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> x e. X )
13		simpl 109	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> D e. ( *Met ` X ) )
14		xmetsym 15469	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
15	13, 12, 11, 14	syl3anc 1278	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) = ( y D x ) )
16	10	simp3d 1042	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) e. RR )
17	15, 16	eqeltrrd 2316	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( y D x ) e. RR )
18	1	xmeterval 15536	. . . 4 |- ( D e. ( *Met ` X ) -> ( y .~ x <-> ( y e. X /\ x e. X /\ ( y D x ) e. RR ) ) )
19	18	adantr 276	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( y .~ x <-> ( y e. X /\ x e. X /\ ( y D x ) e. RR ) ) )
20	11, 12, 17, 19	mpbir3and 1211	. 2 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y .~ x )
21	12	adantrr 483	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x e. X )
22	1	xmeterval 15536	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( y .~ z <-> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) ) )
23	22	biimpa 296	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ y .~ z ) -> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) )
24	23	adantrl 482	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) )
25	24	simp2d 1041	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> z e. X )
26		simpl 109	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> D e. ( *Met ` X ) )
27	16	adantrr 483	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D y ) e. RR )
28	24	simp3d 1042	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y D z ) e. RR )
29		rexadd 10254	. . . . . 6 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) +e ( y D z ) ) = ( ( x D y ) + ( y D z ) ) )
30		readdcl 8305	. . . . . 6 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) + ( y D z ) ) e. RR )
31	29, 30	eqeltrd 2315	. . . . 5 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) +e ( y D z ) ) e. RR )
32	27, 28, 31	syl2anc 415	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( ( x D y ) +e ( y D z ) ) e. RR )
33	11	adantrr 483	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> y e. X )
34		xmettri 15473	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x e. X /\ z e. X /\ y e. X ) ) -> ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) )
35	26, 21, 25, 33, 34	syl13anc 1280	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) )
36		xmetlecl 15468	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x e. X /\ z e. X ) /\ ( ( ( x D y ) +e ( y D z ) ) e. RR /\ ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) ) ) -> ( x D z ) e. RR )
37	26, 21, 25, 32, 35, 36	syl122anc 1287	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) e. RR )
38	1	xmeterval 15536	. . . 4 |- ( D e. ( *Met ` X ) -> ( x .~ z <-> ( x e. X /\ z e. X /\ ( x D z ) e. RR ) ) )
39	38	adantr 276	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x .~ z <-> ( x e. X /\ z e. X /\ ( x D z ) e. RR ) ) )
40	21, 25, 37, 39	mpbir3and 1211	. 2 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x .~ z )
41		xmet0 15464	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x D x ) = 0 )
42		0re 8326	. . . . . . 7 |- 0 e. RR
43	41, 42	eqeltrdi 2329	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x D x ) e. RR )
44	43	ex 115	. . . . 5 |- ( D e. ( *Met ` X ) -> ( x e. X -> ( x D x ) e. RR ) )
45	44	pm4.71rd 398	. . . 4 |- ( D e. ( *Met ` X ) -> ( x e. X <-> ( ( x D x ) e. RR /\ x e. X ) ) )
46		df-3an 1011	. . . . 5 |- ( ( x e. X /\ x e. X /\ ( x D x ) e. RR ) <-> ( ( x e. X /\ x e. X ) /\ ( x D x ) e. RR ) )
47		anidm 400	. . . . . 6 |- ( ( x e. X /\ x e. X ) <-> x e. X )
48	47	anbi2ci 463	. . . . 5 |- ( ( ( x e. X /\ x e. X ) /\ ( x D x ) e. RR ) <-> ( ( x D x ) e. RR /\ x e. X ) )
49	46, 48	bitri 184	. . . 4 |- ( ( x e. X /\ x e. X /\ ( x D x ) e. RR ) <-> ( ( x D x ) e. RR /\ x e. X ) )
50	45, 49	bitr4di 198	. . 3 |- ( D e. ( *Met ` X ) -> ( x e. X <-> ( x e. X /\ x e. X /\ ( x D x ) e. RR ) ) )
51	1	xmeterval 15536	. . 3 |- ( D e. ( *Met ` X ) -> ( x .~ x <-> ( x e. X /\ x e. X /\ ( x D x ) e. RR ) ) )
52	50, 51	bitr4d 191	. 2 |- ( D e. ( *Met ` X ) -> ( x e. X <-> x .~ x ) )
53	8, 20, 40, 52	iserd 6833	1 |- ( D e. ( *Met ` X ) -> .~ Er X )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4130   X. cxp 4772   `'ccnv 4773   dom cdm 4774   "cima 4777   Rel wrel 4779   ` cfv 5377  (class class class)co 6085   Er wer 6804   RRcr 8178   0cc0 8179   + caddc 8182   RR*cxr 8359   <_ cle 8361   +ecxad 10172   *Metcxmet 14873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-er 6807  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-2 9363  df-xadd 10175  df-xmet 14881

This theorem is used by:  blpnfctr  15540  xmetresbl  15541