Theorem blval2 22367

Description: The ball around a point P, alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blval2	|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) = ( ( `' D " ( 0 [,) R ) ) " { P } ) )

Proof of Theorem blval2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpxr 11840	. . 3 |- ( R e. RR+ -> R e. RR* )
2		blvalps 22190	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
3	1, 2	syl3an3 1361	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
4		nfv 1843	. . 3 |- F/ x ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ )
5		nfcv 2764	. . 3 |- F/_ x ( ( `' D " ( 0 [,) R ) ) " { P } )
6		nfrab1 3122	. . 3 |- F/_ x { x e. X | ( P D x ) < R }
7		psmetf 22111	. . . . . . 7 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
8		ffn 6045	. . . . . . 7 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
9		elpreima 6337	. . . . . . 7 |- ( D Fn ( X X. X ) -> ( <. P , x >. e. ( `' D " ( 0 [,) R ) ) <-> ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) ) )
10	7, 8, 9	3syl 18	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( <. P , x >. e. ( `' D " ( 0 [,) R ) ) <-> ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) ) )
11	10	3ad2ant1 1082	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( <. P , x >. e. ( `' D " ( 0 [,) R ) ) <-> ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) ) )
12		opelxp 5146	. . . . . . . . . 10 |- ( <. P , x >. e. ( X X. X ) <-> ( P e. X /\ x e. X ) )
13	12	baib 944	. . . . . . . . 9 |- ( P e. X -> ( <. P , x >. e. ( X X. X ) <-> x e. X ) )
14	13	3ad2ant2 1083	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( <. P , x >. e. ( X X. X ) <-> x e. X ) )
15	14	biimpd 219	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( <. P , x >. e. ( X X. X ) -> x e. X ) )
16	15	adantrd 484	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) -> x e. X ) )
17		simprl 794	. . . . . . 7 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ ( x e. X /\ ( P D x ) < R ) ) -> x e. X )
18	17	ex 450	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( ( x e. X /\ ( P D x ) < R ) -> x e. X ) )
19		simpl2 1065	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> P e. X )
20	19, 13	syl 17	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( <. P , x >. e. ( X X. X ) <-> x e. X ) )
21		df-ov 6653	. . . . . . . . . 10 |- ( P D x ) = ( D ` <. P , x >. )
22	21	eleq1i 2692	. . . . . . . . 9 |- ( ( P D x ) e. ( 0 [,) R ) <-> ( D ` <. P , x >. ) e. ( 0 [,) R ) )
23		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
24		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> R e. RR+ )
25	24	rpxrd 11873	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> R e. RR* )
26		elico1 12218	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( ( P D x ) e. ( 0 [,) R ) <-> ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) /\ ( P D x ) < R ) ) )
27	23, 25, 26	sylancr 695	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( P D x ) e. ( 0 [,) R ) <-> ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) /\ ( P D x ) < R ) ) )
28		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> D e. (PsMet ` X ) )
29		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> x e. X )
30		psmetcl 22112	. . . . . . . . . . . . . 14 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
31	28, 19, 29, 30	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( P D x ) e. RR* )
32		psmetge0 22117	. . . . . . . . . . . . . 14 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
33	28, 19, 29, 32	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> 0 <_ ( P D x ) )
34	31, 33	jca 554	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) ) )
35	34	biantrurd 529	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( P D x ) < R <-> ( ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) ) /\ ( P D x ) < R ) ) )
36		df-3an 1039	. . . . . . . . . . 11 |- ( ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) /\ ( P D x ) < R ) <-> ( ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) ) /\ ( P D x ) < R ) )
37	35, 36	syl6rbbr 279	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( ( P D x ) e. RR* /\ 0 <_ ( P D x ) /\ ( P D x ) < R ) <-> ( P D x ) < R ) )
38	27, 37	bitrd 268	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( P D x ) e. ( 0 [,) R ) <-> ( P D x ) < R ) )
39	22, 38	syl5bbr 274	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( D ` <. P , x >. ) e. ( 0 [,) R ) <-> ( P D x ) < R ) )
40	20, 39	anbi12d 747	. . . . . . 7 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) /\ x e. X ) -> ( ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) <-> ( x e. X /\ ( P D x ) < R ) ) )
41	40	ex 450	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( x e. X -> ( ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) <-> ( x e. X /\ ( P D x ) < R ) ) ) )
42	16, 18, 41	pm5.21ndd 369	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( ( <. P , x >. e. ( X X. X ) /\ ( D ` <. P , x >. ) e. ( 0 [,) R ) ) <-> ( x e. X /\ ( P D x ) < R ) ) )
43	11, 42	bitrd 268	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( <. P , x >. e. ( `' D " ( 0 [,) R ) ) <-> ( x e. X /\ ( P D x ) < R ) ) )
44		vex 3203	. . . . . 6 |- x e. _V
45		elimasng 5491	. . . . . 6 |- ( ( P e. X /\ x e. _V ) -> ( x e. ( ( `' D " ( 0 [,) R ) ) " { P } ) <-> <. P , x >. e. ( `' D " ( 0 [,) R ) ) ) )
46	44, 45	mpan2 707	. . . . 5 |- ( P e. X -> ( x e. ( ( `' D " ( 0 [,) R ) ) " { P } ) <-> <. P , x >. e. ( `' D " ( 0 [,) R ) ) ) )
47	46	3ad2ant2 1083	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( x e. ( ( `' D " ( 0 [,) R ) ) " { P } ) <-> <. P , x >. e. ( `' D " ( 0 [,) R ) ) ) )
48		rabid 3116	. . . . 5 |- ( x e. { x e. X | ( P D x ) < R } <-> ( x e. X /\ ( P D x ) < R ) )
49	48	a1i 11	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( x e. { x e. X | ( P D x ) < R } <-> ( x e. X /\ ( P D x ) < R ) ) )
50	43, 47, 49	3bitr4d 300	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( x e. ( ( `' D " ( 0 [,) R ) ) " { P } ) <-> x e. { x e. X | ( P D x ) < R } ) )
51	4, 5, 6, 50	eqrd 3622	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( ( `' D " ( 0 [,) R ) ) " { P } ) = { x e. X | ( P D x ) < R } )
52	3, 51	eqtr4d 2659	1 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) = ( ( `' D " ( 0 [,) R ) ) " { P } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730   ballcbl 19733

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-ico 12181  df-psmet 19738  df-bl 19741

This theorem is referenced by:  elbl4  22368  metustbl  22371  psmetutop  22372