Theorem metustto 22358

Description: Any two elements of the filter base generated by the metric D can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )

Assertion

Ref	Expression
metustto	|- ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) -> ( A C_ B \/ B C_ A ) )

Distinct variable groups:   B, a   D, a   X, a   A, a   F, a

Proof of Theorem metustto

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 |- ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> a e. RR+ )
2	1	rpred 11872	. . . 4 |- ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> a e. RR )
3		simplr 792	. . . . 5 |- ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> b e. RR+ )
4	3	rpred 11872	. . . 4 |- ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> b e. RR )
5		simpllr 799	. . . . . . . 8 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> b e. RR+ )
6	5	rpred 11872	. . . . . . 7 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> b e. RR )
7		0xr 10086	. . . . . . . . . 10 |- 0 e. RR*
8	7	a1i 11	. . . . . . . . 9 |- ( ( b e. RR /\ a <_ b ) -> 0 e. RR* )
9		simpl 473	. . . . . . . . . 10 |- ( ( b e. RR /\ a <_ b ) -> b e. RR )
10	9	rexrd 10089	. . . . . . . . 9 |- ( ( b e. RR /\ a <_ b ) -> b e. RR* )
11		0le0 11110	. . . . . . . . . 10 |- 0 <_ 0
12	11	a1i 11	. . . . . . . . 9 |- ( ( b e. RR /\ a <_ b ) -> 0 <_ 0 )
13		simpr 477	. . . . . . . . 9 |- ( ( b e. RR /\ a <_ b ) -> a <_ b )
14		icossico 12243	. . . . . . . . 9 |- ( ( ( 0 e. RR* /\ b e. RR* ) /\ ( 0 <_ 0 /\ a <_ b ) ) -> ( 0 [,) a ) C_ ( 0 [,) b ) )
15	8, 10, 12, 13, 14	syl22anc 1327	. . . . . . . 8 |- ( ( b e. RR /\ a <_ b ) -> ( 0 [,) a ) C_ ( 0 [,) b ) )
16		imass2 5501	. . . . . . . 8 |- ( ( 0 [,) a ) C_ ( 0 [,) b ) -> ( `' D " ( 0 [,) a ) ) C_ ( `' D " ( 0 [,) b ) ) )
17	15, 16	syl 17	. . . . . . 7 |- ( ( b e. RR /\ a <_ b ) -> ( `' D " ( 0 [,) a ) ) C_ ( `' D " ( 0 [,) b ) ) )
18	6, 17	sylancom 701	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> ( `' D " ( 0 [,) a ) ) C_ ( `' D " ( 0 [,) b ) ) )
19		simplrl 800	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> A = ( `' D " ( 0 [,) a ) ) )
20		simplrr 801	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> B = ( `' D " ( 0 [,) b ) ) )
21	18, 19, 20	3sstr4d 3648	. . . . 5 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> A C_ B )
22	21	orcd 407	. . . 4 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ a <_ b ) -> ( A C_ B \/ B C_ A ) )
23		simplll 798	. . . . . . . 8 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> a e. RR+ )
24	23	rpred 11872	. . . . . . 7 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> a e. RR )
25	7	a1i 11	. . . . . . . . 9 |- ( ( a e. RR /\ b <_ a ) -> 0 e. RR* )
26		simpl 473	. . . . . . . . . 10 |- ( ( a e. RR /\ b <_ a ) -> a e. RR )
27	26	rexrd 10089	. . . . . . . . 9 |- ( ( a e. RR /\ b <_ a ) -> a e. RR* )
28	11	a1i 11	. . . . . . . . 9 |- ( ( a e. RR /\ b <_ a ) -> 0 <_ 0 )
29		simpr 477	. . . . . . . . 9 |- ( ( a e. RR /\ b <_ a ) -> b <_ a )
30		icossico 12243	. . . . . . . . 9 |- ( ( ( 0 e. RR* /\ a e. RR* ) /\ ( 0 <_ 0 /\ b <_ a ) ) -> ( 0 [,) b ) C_ ( 0 [,) a ) )
31	25, 27, 28, 29, 30	syl22anc 1327	. . . . . . . 8 |- ( ( a e. RR /\ b <_ a ) -> ( 0 [,) b ) C_ ( 0 [,) a ) )
32		imass2 5501	. . . . . . . 8 |- ( ( 0 [,) b ) C_ ( 0 [,) a ) -> ( `' D " ( 0 [,) b ) ) C_ ( `' D " ( 0 [,) a ) ) )
33	31, 32	syl 17	. . . . . . 7 |- ( ( a e. RR /\ b <_ a ) -> ( `' D " ( 0 [,) b ) ) C_ ( `' D " ( 0 [,) a ) ) )
34	24, 33	sylancom 701	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> ( `' D " ( 0 [,) b ) ) C_ ( `' D " ( 0 [,) a ) ) )
35		simplrr 801	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> B = ( `' D " ( 0 [,) b ) ) )
36		simplrl 800	. . . . . 6 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> A = ( `' D " ( 0 [,) a ) ) )
37	34, 35, 36	3sstr4d 3648	. . . . 5 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> B C_ A )
38	37	olcd 408	. . . 4 |- ( ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) /\ b <_ a ) -> ( A C_ B \/ B C_ A ) )
39	2, 4, 22, 38	lecasei 10143	. . 3 |- ( ( ( a e. RR+ /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> ( A C_ B \/ B C_ A ) )
40	39	adantlll 754	. 2 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) /\ a e. RR+ ) /\ b e. RR+ ) /\ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) ) -> ( A C_ B \/ B C_ A ) )
41		metust.1	. . . . . 6 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
42	41	metustel 22355	. . . . 5 |- ( D e. (PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
43	42	biimpa 501	. . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
44	43	3adant3 1081	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
45		oveq2 6658	. . . . . . . . . 10 |- ( a = b -> ( 0 [,) a ) = ( 0 [,) b ) )
46	45	imaeq2d 5466	. . . . . . . . 9 |- ( a = b -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) b ) ) )
47	46	cbvmptv 4750	. . . . . . . 8 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
48	47	rneqi 5352	. . . . . . 7 |- ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
49	41, 48	eqtri 2644	. . . . . 6 |- F = ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
50	49	metustel 22355	. . . . 5 |- ( D e. (PsMet ` X ) -> ( B e. F <-> E. b e. RR+ B = ( `' D " ( 0 [,) b ) ) ) )
51	50	biimpa 501	. . . 4 |- ( ( D e. (PsMet ` X ) /\ B e. F ) -> E. b e. RR+ B = ( `' D " ( 0 [,) b ) ) )
52	51	3adant2 1080	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) -> E. b e. RR+ B = ( `' D " ( 0 [,) b ) ) )
53		reeanv 3107	. . 3 |- ( E. a e. RR+ E. b e. RR+ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) <-> ( E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) /\ E. b e. RR+ B = ( `' D " ( 0 [,) b ) ) ) )
54	44, 52, 53	sylanbrc 698	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) -> E. a e. RR+ E. b e. RR+ ( A = ( `' D " ( 0 [,) a ) ) /\ B = ( `' D " ( 0 [,) b ) ) ) )
55	40, 54	r19.29vva 3081	1 |- ( ( D e. (PsMet ` X ) /\ A e. F /\ B e. F ) -> ( A C_ B \/ B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   RR*cxr 10073   <_ cle 10075   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730

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-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  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-rp 11833  df-ico 12181

This theorem is referenced by:  metustfbas  22362