Theorem xmeter 22238

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 5485	. . . . 5 |- ( `' D " RR ) C_ dom D
3	1, 2	eqsstri 3635	. . . 4 |- .~ C_ dom D
4		xmetf 22134	. . . . 5 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
5		fdm 6051	. . . . 5 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
6	4, 5	syl 17	. . . 4 |- ( D e. ( *Met ` X ) -> dom D = ( X X. X ) )
7	3, 6	syl5sseq 3653	. . 3 |- ( D e. ( *Met ` X ) -> .~ C_ ( X X. X ) )
8		relxp 5227	. . 3 |- Rel ( X X. X )
9		relss 5206	. . 3 |- ( .~ C_ ( X X. X ) -> ( Rel ( X X. X ) -> Rel .~ ) )
10	7, 8, 9	mpisyl 21	. 2 |- ( D e. ( *Met ` X ) -> Rel .~ )
11	1	xmeterval 22237	. . . . 5 |- ( D e. ( *Met ` X ) -> ( x .~ y <-> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) ) )
12	11	biimpa 501	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) )
13	12	simp2d 1074	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y e. X )
14	12	simp1d 1073	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> x e. X )
15		simpl 473	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> D e. ( *Met ` X ) )
16		xmetsym 22152	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
17	15, 14, 13, 16	syl3anc 1326	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) = ( y D x ) )
18	12	simp3d 1075	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( x D y ) e. RR )
19	17, 18	eqeltrrd 2702	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( y D x ) e. RR )
20	1	xmeterval 22237	. . . 4 |- ( D e. ( *Met ` X ) -> ( y .~ x <-> ( y e. X /\ x e. X /\ ( y D x ) e. RR ) ) )
21	20	adantr 481	. . 3 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> ( y .~ x <-> ( y e. X /\ x e. X /\ ( y D x ) e. RR ) ) )
22	13, 14, 19, 21	mpbir3and 1245	. 2 |- ( ( D e. ( *Met ` X ) /\ x .~ y ) -> y .~ x )
23	14	adantrr 753	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x e. X )
24	1	xmeterval 22237	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( y .~ z <-> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) ) )
25	24	biimpa 501	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ y .~ z ) -> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) )
26	25	adantrl 752	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y e. X /\ z e. X /\ ( y D z ) e. RR ) )
27	26	simp2d 1074	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> z e. X )
28		simpl 473	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> D e. ( *Met ` X ) )
29	18	adantrr 753	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D y ) e. RR )
30	26	simp3d 1075	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( y D z ) e. RR )
31		rexadd 12063	. . . . . 6 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) +e ( y D z ) ) = ( ( x D y ) + ( y D z ) ) )
32		readdcl 10019	. . . . . 6 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) + ( y D z ) ) e. RR )
33	31, 32	eqeltrd 2701	. . . . 5 |- ( ( ( x D y ) e. RR /\ ( y D z ) e. RR ) -> ( ( x D y ) +e ( y D z ) ) e. RR )
34	29, 30, 33	syl2anc 693	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( ( x D y ) +e ( y D z ) ) e. RR )
35	13	adantrr 753	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> y e. X )
36		xmettri 22156	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( x e. X /\ z e. X /\ y e. X ) ) -> ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) )
37	28, 23, 27, 35, 36	syl13anc 1328	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) <_ ( ( x D y ) +e ( y D z ) ) )
38		xmetlecl 22151	. . . 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 )
39	28, 23, 27, 34, 37, 38	syl122anc 1335	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x D z ) e. RR )
40	1	xmeterval 22237	. . . 4 |- ( D e. ( *Met ` X ) -> ( x .~ z <-> ( x e. X /\ z e. X /\ ( x D z ) e. RR ) ) )
41	40	adantr 481	. . 3 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> ( x .~ z <-> ( x e. X /\ z e. X /\ ( x D z ) e. RR ) ) )
42	23, 27, 39, 41	mpbir3and 1245	. 2 |- ( ( D e. ( *Met ` X ) /\ ( x .~ y /\ y .~ z ) ) -> x .~ z )
43		xmet0 22147	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x D x ) = 0 )
44		0re 10040	. . . . . . 7 |- 0 e. RR
45	43, 44	syl6eqel 2709	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x D x ) e. RR )
46	45	ex 450	. . . . 5 |- ( D e. ( *Met ` X ) -> ( x e. X -> ( x D x ) e. RR ) )
47	46	pm4.71rd 667	. . . 4 |- ( D e. ( *Met ` X ) -> ( x e. X <-> ( ( x D x ) e. RR /\ x e. X ) ) )
48		df-3an 1039	. . . . 5 |- ( ( x e. X /\ x e. X /\ ( x D x ) e. RR ) <-> ( ( x e. X /\ x e. X ) /\ ( x D x ) e. RR ) )
49		anidm 676	. . . . . 6 |- ( ( x e. X /\ x e. X ) <-> x e. X )
50	49	anbi2ci 732	. . . . 5 |- ( ( ( x e. X /\ x e. X ) /\ ( x D x ) e. RR ) <-> ( ( x D x ) e. RR /\ x e. X ) )
51	48, 50	bitri 264	. . . 4 |- ( ( x e. X /\ x e. X /\ ( x D x ) e. RR ) <-> ( ( x D x ) e. RR /\ x e. X ) )
52	47, 51	syl6bbr 278	. . 3 |- ( D e. ( *Met ` X ) -> ( x e. X <-> ( x e. X /\ x e. X /\ ( x D x ) e. RR ) ) )
53	1	xmeterval 22237	. . 3 |- ( D e. ( *Met ` X ) -> ( x .~ x <-> ( x e. X /\ x e. X /\ ( x D x ) e. RR ) ) )
54	52, 53	bitr4d 271	. 2 |- ( D e. ( *Met ` X ) -> ( x e. X <-> x .~ x ) )
55	10, 22, 42, 54	iserd 7768	1 |- ( D e. ( *Met ` X ) -> .~ Er X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   Er wer 7739   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   <_ cle 10075   +ecxad 11944   *Metcxmt 19731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-xmet 19739

This theorem is referenced by:  blpnfctr  22241  xmetresbl  22242