Theorem bdxmet 12429

Description: The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)

Hypothesis

Ref	Expression
stdbdmet.1	|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )

Assertion

Ref	Expression
bdxmet	|- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )

Distinct variable groups:   x, y, C   x, R, y   x, X, y

Allowed substitution hints:   D( x, y)

Proof of Theorem bdxmet

Dummy variables a b z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 949	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
2		xmetcl 12280	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR* )
3		xmetge0 12293	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x C y ) )
4		elxrge0 9602	. . . . . . 7 |- ( ( x C y ) e. ( 0 [,] +oo ) <-> ( ( x C y ) e. RR* /\ 0 <_ ( x C y ) ) )
5	2, 3, 4	sylanbrc 411	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. ( 0 [,] +oo ) )
6	5	3expb 1150	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
7	1, 6	sylan 279	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
8		xmetf 12278	. . . . . . 7 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
9	8	3ad2ant1 970	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
10	9	ffnd 5209	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C Fn ( X X. X ) )
11		fnovim 5811	. . . . 5 |- ( C Fn ( X X. X ) -> C = ( x e. X , y e. X |-> ( x C y ) ) )
12	10, 11	syl 14	. . . 4 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C = ( x e. X , y e. X |-> ( x C y ) ) )
13		eqidd 2101	. . . 4 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) = ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) )
14		preq1 3547	. . . . 5 |- ( z = ( x C y ) -> { z , R } = { ( x C y ) , R } )
15	14	infeq1d 6814	. . . 4 |- ( z = ( x C y ) -> inf ( { z , R } , RR* , < ) = inf ( { ( x C y ) , R } , RR* , < ) )
16	7, 12, 13, 15	fmpoco 6043	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) ) )
17		stdbdmet.1	. . 3 |- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )
18	16, 17	syl6eqr 2150	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) = D )
19		elxrge0 9602	. . . . . 6 |- ( z e. ( 0 [,] +oo ) <-> ( z e. RR* /\ 0 <_ z ) )
20	19	simplbi 270	. . . . 5 |- ( z e. ( 0 [,] +oo ) -> z e. RR* )
21		simp2 950	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> R e. RR* )
22		xrmincl 10874	. . . . 5 |- ( ( z e. RR* /\ R e. RR* ) -> inf ( { z , R } , RR* , < ) e. RR* )
23	20, 21, 22	syl2anr 286	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ z e. ( 0 [,] +oo ) ) -> inf ( { z , R } , RR* , < ) e. RR* )
24	23	fmpttd 5507	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) : ( 0 [,] +oo ) --> RR* )
25		eqid 2100	. . . . . 6 |- ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) = ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) )
26		preq1 3547	. . . . . . 7 |- ( z = a -> { z , R } = { a , R } )
27	26	infeq1d 6814	. . . . . 6 |- ( z = a -> inf ( { z , R } , RR* , < ) = inf ( { a , R } , RR* , < ) )
28		simpr 109	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> a e. ( 0 [,] +oo ) )
29		elxrge0 9602	. . . . . . . 8 |- ( a e. ( 0 [,] +oo ) <-> ( a e. RR* /\ 0 <_ a ) )
30	29	simplbi 270	. . . . . . 7 |- ( a e. ( 0 [,] +oo ) -> a e. RR* )
31		xrmincl 10874	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) e. RR* )
32	30, 21, 31	syl2anr 286	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> inf ( { a , R } , RR* , < ) e. RR* )
33	25, 27, 28, 32	fvmptd3 5446	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) = inf ( { a , R } , RR* , < ) )
34	33	eqeq1d 2108	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) = 0 <-> inf ( { a , R } , RR* , < ) = 0 ) )
35		0xr 7684	. . . . . . . . 9 |- 0 e. RR*
36	35	a1i 9	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 e. RR* )
37	30	adantl 273	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> a e. RR* )
38	21	adantr 272	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> R e. RR* )
39		xrltmininf 10878	. . . . . . . 8 |- ( ( 0 e. RR* /\ a e. RR* /\ R e. RR* ) -> ( 0 < inf ( { a , R } , RR* , < ) <-> ( 0 < a /\ 0 < R ) ) )
40	36, 37, 38, 39	syl3anc 1184	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 < inf ( { a , R } , RR* , < ) <-> ( 0 < a /\ 0 < R ) ) )
41		simp3 951	. . . . . . . . 9 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> 0 < R )
42	41	adantr 272	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 < R )
43	42	biantrud 300	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 < a <-> ( 0 < a /\ 0 < R ) ) )
44	40, 43	bitr4d 190	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 < inf ( { a , R } , RR* , < ) <-> 0 < a ) )
45	44	notbid 633	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < inf ( { a , R } , RR* , < ) <-> -. 0 < a ) )
46	28, 29	sylib 121	. . . . . . . . . 10 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( a e. RR* /\ 0 <_ a ) )
47	46	simprd 113	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ a )
48		xrltle 9425	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( 0 < R -> 0 <_ R ) )
49	35, 21, 48	sylancr 408	. . . . . . . . . . 11 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( 0 < R -> 0 <_ R ) )
50	41, 49	mpd 13	. . . . . . . . . 10 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> 0 <_ R )
51	50	adantr 272	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ R )
52		xrlemininf 10879	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ a e. RR* /\ R e. RR* ) -> ( 0 <_ inf ( { a , R } , RR* , < ) <-> ( 0 <_ a /\ 0 <_ R ) ) )
53	36, 37, 38, 52	syl3anc 1184	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 <_ inf ( { a , R } , RR* , < ) <-> ( 0 <_ a /\ 0 <_ R ) ) )
54	47, 51, 53	mpbir2and 896	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ inf ( { a , R } , RR* , < ) )
55		xrlenlt 7701	. . . . . . . . 9 |- ( ( 0 e. RR* /\ inf ( { a , R } , RR* , < ) e. RR* ) -> ( 0 <_ inf ( { a , R } , RR* , < ) <-> -. inf ( { a , R } , RR* , < ) < 0 ) )
56	35, 32, 55	sylancr 408	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 <_ inf ( { a , R } , RR* , < ) <-> -. inf ( { a , R } , RR* , < ) < 0 ) )
57	54, 56	mpbid 146	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> -. inf ( { a , R } , RR* , < ) < 0 )
58	57	biantrurd 301	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < inf ( { a , R } , RR* , < ) <-> ( -. inf ( { a , R } , RR* , < ) < 0 /\ -. 0 < inf ( { a , R } , RR* , < ) ) ) )
59		xrlttri3 9424	. . . . . . 7 |- ( (inf ( { a , R } , RR* , < ) e. RR* /\ 0 e. RR* ) -> (inf ( { a , R } , RR* , < ) = 0 <-> ( -. inf ( { a , R } , RR* , < ) < 0 /\ -. 0 < inf ( { a , R } , RR* , < ) ) ) )
60	32, 36, 59	syl2anc 406	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> (inf ( { a , R } , RR* , < ) = 0 <-> ( -. inf ( { a , R } , RR* , < ) < 0 /\ -. 0 < inf ( { a , R } , RR* , < ) ) ) )
61	58, 60	bitr4d 190	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < inf ( { a , R } , RR* , < ) <-> inf ( { a , R } , RR* , < ) = 0 ) )
62		xrlenlt 7701	. . . . . . . . 9 |- ( ( 0 e. RR* /\ a e. RR* ) -> ( 0 <_ a <-> -. a < 0 ) )
63	35, 37, 62	sylancr 408	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 <_ a <-> -. a < 0 ) )
64	47, 63	mpbid 146	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> -. a < 0 )
65	64	biantrurd 301	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < a <-> ( -. a < 0 /\ -. 0 < a ) ) )
66		xrlttri3 9424	. . . . . . 7 |- ( ( a e. RR* /\ 0 e. RR* ) -> ( a = 0 <-> ( -. a < 0 /\ -. 0 < a ) ) )
67	37, 36, 66	syl2anc 406	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( a = 0 <-> ( -. a < 0 /\ -. 0 < a ) ) )
68	65, 67	bitr4d 190	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < a <-> a = 0 ) )
69	45, 61, 68	3bitr3d 217	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> (inf ( { a , R } , RR* , < ) = 0 <-> a = 0 ) )
70	34, 69	bitrd 187	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) = 0 <-> a = 0 ) )
71	30	ad2antrl 477	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> a e. RR* )
72	21	adantr 272	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> R e. RR* )
73		xrmin1inf 10875	. . . . . . . 8 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ a )
74	71, 72, 73	syl2anc 406	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { a , R } , RR* , < ) <_ a )
75	71, 72, 31	syl2anc 406	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { a , R } , RR* , < ) e. RR* )
76		elxrge0 9602	. . . . . . . . . 10 |- ( b e. ( 0 [,] +oo ) <-> ( b e. RR* /\ 0 <_ b ) )
77	76	simplbi 270	. . . . . . . . 9 |- ( b e. ( 0 [,] +oo ) -> b e. RR* )
78	77	ad2antll 478	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. RR* )
79		xrletr 9432	. . . . . . . 8 |- ( (inf ( { a , R } , RR* , < ) e. RR* /\ a e. RR* /\ b e. RR* ) -> ( (inf ( { a , R } , RR* , < ) <_ a /\ a <_ b ) -> inf ( { a , R } , RR* , < ) <_ b ) )
80	75, 71, 78, 79	syl3anc 1184	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( (inf ( { a , R } , RR* , < ) <_ a /\ a <_ b ) -> inf ( { a , R } , RR* , < ) <_ b ) )
81	74, 80	mpand 423	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a <_ b -> inf ( { a , R } , RR* , < ) <_ b ) )
82		xrmin2inf 10876	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ R )
83	71, 72, 82	syl2anc 406	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { a , R } , RR* , < ) <_ R )
84	81, 83	jctird 313	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a <_ b -> (inf ( { a , R } , RR* , < ) <_ b /\ inf ( { a , R } , RR* , < ) <_ R ) ) )
85		xrlemininf 10879	. . . . . 6 |- ( (inf ( { a , R } , RR* , < ) e. RR* /\ b e. RR* /\ R e. RR* ) -> (inf ( { a , R } , RR* , < ) <_ inf ( { b , R } , RR* , < ) <-> (inf ( { a , R } , RR* , < ) <_ b /\ inf ( { a , R } , RR* , < ) <_ R ) ) )
86	75, 78, 72, 85	syl3anc 1184	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> (inf ( { a , R } , RR* , < ) <_ inf ( { b , R } , RR* , < ) <-> (inf ( { a , R } , RR* , < ) <_ b /\ inf ( { a , R } , RR* , < ) <_ R ) ) )
87	84, 86	sylibrd 168	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a <_ b -> inf ( { a , R } , RR* , < ) <_ inf ( { b , R } , RR* , < ) ) )
88	33	adantrr 466	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) = inf ( { a , R } , RR* , < ) )
89		preq1 3547	. . . . . . 7 |- ( z = b -> { z , R } = { b , R } )
90	89	infeq1d 6814	. . . . . 6 |- ( z = b -> inf ( { z , R } , RR* , < ) = inf ( { b , R } , RR* , < ) )
91		simpr 109	. . . . . . 7 |- ( ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) -> b e. ( 0 [,] +oo ) )
92	91	adantl 273	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. ( 0 [,] +oo ) )
93		xrmincl 10874	. . . . . . 7 |- ( ( b e. RR* /\ R e. RR* ) -> inf ( { b , R } , RR* , < ) e. RR* )
94	78, 72, 93	syl2anc 406	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { b , R } , RR* , < ) e. RR* )
95	25, 90, 92, 94	fvmptd3 5446	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` b ) = inf ( { b , R } , RR* , < ) )
96	88, 95	breq12d 3888	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) <_ ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` b ) <-> inf ( { a , R } , RR* , < ) <_ inf ( { b , R } , RR* , < ) ) )
97	87, 96	sylibrd 168	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a <_ b -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) <_ ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` b ) ) )
98	29	simprbi 271	. . . . . 6 |- ( a e. ( 0 [,] +oo ) -> 0 <_ a )
99	98	ad2antrl 477	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ a )
100	76	simprbi 271	. . . . . 6 |- ( b e. ( 0 [,] +oo ) -> 0 <_ b )
101	100	ad2antll 478	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ b )
102	41	adantr 272	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 < R )
103		xrbdtri 10884	. . . . 5 |- ( ( ( a e. RR* /\ 0 <_ a ) /\ ( b e. RR* /\ 0 <_ b ) /\ ( R e. RR* /\ 0 < R ) ) -> inf ( { ( a +e b ) , R } , RR* , < ) <_ (inf ( { a , R } , RR* , < ) +einf ( { b , R } , RR* , < ) ) )
104	71, 99, 78, 101, 72, 102, 103	syl222anc 1200	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { ( a +e b ) , R } , RR* , < ) <_ (inf ( { a , R } , RR* , < ) +einf ( { b , R } , RR* , < ) ) )
105		preq1 3547	. . . . . 6 |- ( z = ( a +e b ) -> { z , R } = { ( a +e b ) , R } )
106	105	infeq1d 6814	. . . . 5 |- ( z = ( a +e b ) -> inf ( { z , R } , RR* , < ) = inf ( { ( a +e b ) , R } , RR* , < ) )
107		ge0xaddcl 9607	. . . . . 6 |- ( ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) -> ( a +e b ) e. ( 0 [,] +oo ) )
108	107	adantl 273	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a +e b ) e. ( 0 [,] +oo ) )
109	71, 78	xaddcld 9508	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( a +e b ) e. RR* )
110		xrmincl 10874	. . . . . 6 |- ( ( ( a +e b ) e. RR* /\ R e. RR* ) -> inf ( { ( a +e b ) , R } , RR* , < ) e. RR* )
111	109, 72, 110	syl2anc 406	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> inf ( { ( a +e b ) , R } , RR* , < ) e. RR* )
112	25, 106, 108, 111	fvmptd3 5446	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` ( a +e b ) ) = inf ( { ( a +e b ) , R } , RR* , < ) )
113	88, 95	oveq12d 5724	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) +e ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` b ) ) = (inf ( { a , R } , RR* , < ) +einf ( { b , R } , RR* , < ) ) )
114	104, 112, 113	3brtr4d 3905	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` ( a +e b ) ) <_ ( ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` a ) +e ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) ` b ) ) )
115	1, 24, 70, 97, 114	comet 12427	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) e. ( *Met ` X ) )
116	18, 115	eqeltrrd 2177	1 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   {cpr 3475   class class class wbr 3875   |-> cmpt 3929   X. cxp 4475   o. ccom 4481   Fn wfn 5054   -->wf 5055   ` cfv 5059  (class class class)co 5706   e. cmpo 5708  infcinf 6785   0cc0 7500   +oocpnf 7669   RR*cxr 7671   < clt 7672   <_ cle 7673   +ecxad 9398   [,]cicc 9515   *Metcxmet 11931

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-xneg 9400  df-xadd 9401  df-icc 9519  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-xmet 11939

This theorem is referenced by:  bdmet  12430  bdbl  12431  bdmopn  12432