Theorem metustid 21647

Description: The identity diagonal is included in all elements of the filter base generated by the metric D. (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
metustid	|- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A )

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

Proof of Theorem metustid

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5138	. . 3 |- Rel ( _I |` X )
2	1	a1i 11	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> Rel ( _I |` X ) )
3		vex 3034	. . . . . . . . . . . . . . 15 |- q e. _V
4	3	brres 5117	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> ( p _I q /\ p e. X ) )
5		df-br 4396	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> <. p , q >. e. ( _I |` X ) )
6	3	ideq 4992	. . . . . . . . . . . . . . 15 |- ( p _I q <-> p = q )
7	6	anbi1i 709	. . . . . . . . . . . . . 14 |- ( ( p _I q /\ p e. X ) <-> ( p = q /\ p e. X ) )
8	4, 5, 7	3bitr3i 283	. . . . . . . . . . . . 13 |- ( <. p , q >. e. ( _I |` X ) <-> ( p = q /\ p e. X ) )
9	8	biimpi 199	. . . . . . . . . . . 12 |- ( <. p , q >. e. ( _I |` X ) -> ( p = q /\ p e. X ) )
10	9	ad2antlr 741	. . . . . . . . . . 11 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p = q /\ p e. X ) )
11	10	simpld 466	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p = q )
12		df-ov 6311	. . . . . . . . . . 11 |- ( p D p ) = ( D ` <. p , p >. )
13		opeq2 4159	. . . . . . . . . . . 12 |- ( p = q -> <. p , p >. = <. p , q >. )
14	13	fveq2d 5883	. . . . . . . . . . 11 |- ( p = q -> ( D ` <. p , p >. ) = ( D ` <. p , q >. ) )
15	12, 14	syl5eq 2517	. . . . . . . . . 10 |- ( p = q -> ( p D p ) = ( D ` <. p , q >. ) )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = ( D ` <. p , q >. ) )
17		simplll 776	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> D e. (PsMet ` X ) )
18	10	simprd 470	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p e. X )
19		psmet0 21402	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ p e. X ) -> ( p D p ) = 0 )
20	17, 18, 19	syl2anc 673	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = 0 )
21	16, 20	eqtr3d 2507	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) = 0 )
22		0xr 9705	. . . . . . . . . . 11 |- 0 e. RR*
23	22	a1i 11	. . . . . . . . . 10 |- ( a e. RR+ -> 0 e. RR* )
24		rpxr 11332	. . . . . . . . . 10 |- ( a e. RR+ -> a e. RR* )
25		rpgt0 11336	. . . . . . . . . 10 |- ( a e. RR+ -> 0 < a )
26		lbico1 11714	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ a e. RR* /\ 0 < a ) -> 0 e. ( 0 [,) a ) )
27	23, 24, 25, 26	syl3anc 1292	. . . . . . . . 9 |- ( a e. RR+ -> 0 e. ( 0 [,) a ) )
28	27	adantl 473	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> 0 e. ( 0 [,) a ) )
29	21, 28	eqeltrd 2549	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) )
30		psmetf 21400	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
31		ffun 5742	. . . . . . . . . 10 |- ( D : ( X X. X ) --> RR* -> Fun D )
32	30, 31	syl 17	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> Fun D )
33	32	ad3antrrr 744	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> Fun D )
34	11, 18	eqeltrrd 2550	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> q e. X )
35		opelxpi 4871	. . . . . . . . . 10 |- ( ( p e. X /\ q e. X ) -> <. p , q >. e. ( X X. X ) )
36	18, 34, 35	syl2anc 673	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( X X. X ) )
37		fdm 5745	. . . . . . . . . . 11 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
38	30, 37	syl 17	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
39	38	ad3antrrr 744	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> dom D = ( X X. X ) )
40	36, 39	eleqtrrd 2552	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. dom D )
41		fvimacnv 6012	. . . . . . . 8 |- ( ( Fun D /\ <. p , q >. e. dom D ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
42	33, 40, 41	syl2anc 673	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
43	29, 42	mpbid 215	. . . . . 6 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
44	43	adantr 472	. . . . 5 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
45		simpr 468	. . . . 5 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> A = ( `' D " ( 0 [,) a ) ) )
46	44, 45	eleqtrrd 2552	. . . 4 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. A )
47		simplr 770	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> A e. F )
48		metust.1	. . . . . . 7 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
49	48	metustel 21643	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
50	49	ad2antrr 740	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
51	47, 50	mpbid 215	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
52	46, 51	r19.29a 2918	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> <. p , q >. e. A )
53	52	ex 441	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( <. p , q >. e. ( _I |` X ) -> <. p , q >. e. A ) )
54	2, 53	relssdv 4932	1 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   E.wrex 2757   C_ wss 3390   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   _I cid 4749   X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   Rel wrel 4844   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR*cxr 9692   < clt 9693   RR+crp 11325   [,)cico 11662  PsMetcpsmet 19031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-rp 11326  df-ico 11666  df-psmet 19039

This theorem is referenced by:  metustfbas  21650  metust  21651