Theorem xmeter 15150

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 5097	. . . . 5 |- ( `' D " RR ) C_ dom D
3	1, 2	eqsstri 3257	. . . 4 |- .~ C_ dom D
4		xmetf 15064	. . . 4 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
5	3, 4	fssdm 5494	. . 3 |- ( D e. ( *Met ` X ) -> .~ C_ ( X X. X ) )
6		relxp 4833	. . 3 |- Rel ( X X. X )
7		relss 4811	. . 3 |- ( .~ C_ ( X X. X ) -> ( Rel ( X X. X ) -> Rel .~ ) )
8	5, 6, 7	mpisyl 1489	. 2 |- ( D e. ( *Met ` X ) -> Rel .~ )
9	1	xmeterval 15149	. . . . 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 1034	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y e. X )
12	10	simp1d 1033	. . 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 15082	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
15	13, 12, 11, 14	syl3anc 1271	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) = ( y D x ) )
16	10	simp3d 1035	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) e. RR )
17	15, 16	eqeltrrd 2307	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( y D x ) e. RR )
18	1	xmeterval 15149	. . . 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 1204	. 2 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y .~ x )
21	12	adantrr 479	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x e. X )
22	1	xmeterval 15149	. . . . . 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 478	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) )
25	24	simp2d 1034	. . 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 479	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D y ) e. RR )
28	24	simp3d 1035	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y D z ) e. RR )
29		rexadd 10077	. . . . . 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 8148	. . . . . 6 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) + ( y D z ) ) e. RR )
31	29, 30	eqeltrd 2306	. . . . 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 411	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( ( x D y ) +e ( y D z ) ) e. RR )
33	11	adantrr 479	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> y e. X )
34		xmettri 15086	. . . . 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 1273	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) )
36		xmetlecl 15081	. . . 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 1280	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) e. RR )
38	1	xmeterval 15149	. . . 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 1204	. 2 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x .~ z )
41		xmet0 15077	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x D x ) = 0 )
42		0re 8169	. . . . . . 7 |- 0 e. RR
43	41, 42	eqeltrdi 2320	. . . . . 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 394	. . . 4 |- ( D e. ( *Met ` X ) -> ( x e. X <-> ( ( x D x ) e. RR /\ x e. X ) ) )
46		df-3an 1004	. . . . 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 396	. . . . . 6 |- ( ( x e. X /\ x e. X ) <-> x e. X )
48	47	anbi2ci 459	. . . . 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 15149	. . 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 6723	1 |- ( D e. ( *Met ` X ) -> .~ Er X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3198   class class class wbr 4086   X. cxp 4721   `'ccnv 4722   dom cdm 4723   "cima 4726   Rel wrel 4728   ` cfv 5324  (class class class)co 6013   Er wer 6694   RRcr 8021   0cc0 8022   + caddc 8025   RR*cxr 8203   <_ cle 8205   +ecxad 9995   *Metcxmet 14540

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-er 6697  df-map 6814  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-2 9192  df-xadd 9998  df-xmet 14548

This theorem is referenced by:  blpnfctr  15153  xmetresbl  15154