Theorem bdxmet 15088

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 1000	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
2		xmetcl 14939	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR* )
3		xmetge0 14952	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x C y ) )
4		elxrge0 10135	. . . . . . 7 |- ( ( x C y ) e. ( 0 [,] +oo ) <-> ( ( x C y ) e. RR* /\ 0 <_ ( x C y ) ) )
5	2, 3, 4	sylanbrc 417	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. ( 0 [,] +oo ) )
6	5	3expb 1207	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
7	1, 6	sylan 283	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
8		xmetf 14937	. . . . . . 7 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
9	8	3ad2ant1 1021	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
10	9	ffnd 5446	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C Fn ( X X. X ) )
11		fnovim 6077	. . . . 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 2208	. . . 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 3720	. . . . 5 |- ( z = ( x C y ) -> { z , R } = { ( x C y ) , R } )
15	14	infeq1d 7140	. . . 4 |- ( z = ( x C y ) -> inf ( { z , R } , RR* , < ) = inf ( { ( x C y ) , R } , RR* , < ) )
16	7, 12, 13, 15	fmpoco 6325	. . 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	eqtr4di 2258	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) = D )
19		elxrge0 10135	. . . . . 6 |- ( z e. ( 0 [,] +oo ) <-> ( z e. RR* /\ 0 <_ z ) )
20	19	simplbi 274	. . . . 5 |- ( z e. ( 0 [,] +oo ) -> z e. RR* )
21		simp2 1001	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> R e. RR* )
22		xrmincl 11692	. . . . 5 |- ( ( z e. RR* /\ R e. RR* ) -> inf ( { z , R } , RR* , < ) e. RR* )
23	20, 21, 22	syl2anr 290	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ z e. ( 0 [,] +oo ) ) -> inf ( { z , R } , RR* , < ) e. RR* )
24	23	fmpttd 5758	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) : ( 0 [,] +oo ) --> RR* )
25		eqid 2207	. . . . . 6 |- ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) = ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) )
26		preq1 3720	. . . . . . 7 |- ( z = a -> { z , R } = { a , R } )
27	26	infeq1d 7140	. . . . . 6 |- ( z = a -> inf ( { z , R } , RR* , < ) = inf ( { a , R } , RR* , < ) )
28		simpr 110	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> a e. ( 0 [,] +oo ) )
29		elxrge0 10135	. . . . . . . 8 |- ( a e. ( 0 [,] +oo ) <-> ( a e. RR* /\ 0 <_ a ) )
30	29	simplbi 274	. . . . . . 7 |- ( a e. ( 0 [,] +oo ) -> a e. RR* )
31		xrmincl 11692	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) e. RR* )
32	30, 21, 31	syl2anr 290	. . . . . 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 5696	. . . . 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 2216	. . . 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 8154	. . . . . . . . 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 277	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> a e. RR* )
38	21	adantr 276	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> R e. RR* )
39		xrltmininf 11696	. . . . . . . 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 1250	. . . . . . 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 1002	. . . . . . . . 9 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> 0 < R )
42	41	adantr 276	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 < R )
43	42	biantrud 304	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 < a <-> ( 0 < a /\ 0 < R ) ) )
44	40, 43	bitr4d 191	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 < inf ( { a , R } , RR* , < ) <-> 0 < a ) )
45	44	notbid 669	. . . . 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 122	. . . . . . . . . 10 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( a e. RR* /\ 0 <_ a ) )
47	46	simprd 114	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ a )
48		xrltle 9955	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( 0 < R -> 0 <_ R ) )
49	35, 21, 48	sylancr 414	. . . . . . . . . . 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 276	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ R )
52		xrlemininf 11697	. . . . . . . . . 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 1250	. . . . . . . . 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 947	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ inf ( { a , R } , RR* , < ) )
55		xrlenlt 8172	. . . . . . . . 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 414	. . . . . . . 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 147	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> -. inf ( { a , R } , RR* , < ) < 0 )
58	57	biantrurd 305	. . . . . 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 9954	. . . . . . 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 411	. . . . . 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 191	. . . . 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 8172	. . . . . . . . 9 |- ( ( 0 e. RR* /\ a e. RR* ) -> ( 0 <_ a <-> -. a < 0 ) )
63	35, 37, 62	sylancr 414	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( 0 <_ a <-> -. a < 0 ) )
64	47, 63	mpbid 147	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> -. a < 0 )
65	64	biantrurd 305	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < a <-> ( -. a < 0 /\ -. 0 < a ) ) )
66		xrlttri3 9954	. . . . . . 7 |- ( ( a e. RR* /\ 0 e. RR* ) -> ( a = 0 <-> ( -. a < 0 /\ -. 0 < a ) ) )
67	37, 36, 66	syl2anc 411	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( a = 0 <-> ( -. a < 0 /\ -. 0 < a ) ) )
68	65, 67	bitr4d 191	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < a <-> a = 0 ) )
69	45, 61, 68	3bitr3d 218	. . . 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 188	. . 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 490	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> a e. RR* )
72	21	adantr 276	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> R e. RR* )
73		xrmin1inf 11693	. . . . . . . 8 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ a )
74	71, 72, 73	syl2anc 411	. . . . . . 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 411	. . . . . . . 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 10135	. . . . . . . . . 10 |- ( b e. ( 0 [,] +oo ) <-> ( b e. RR* /\ 0 <_ b ) )
77	76	simplbi 274	. . . . . . . . 9 |- ( b e. ( 0 [,] +oo ) -> b e. RR* )
78	77	ad2antll 491	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. RR* )
79		xrletr 9965	. . . . . . . 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 1250	. . . . . . 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 429	. . . . . 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 11694	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ R )
83	71, 72, 82	syl2anc 411	. . . . . 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 317	. . . . 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 11697	. . . . . 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 1250	. . . . 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 169	. . . 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 479	. . . . 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 3720	. . . . . . 7 |- ( z = b -> { z , R } = { b , R } )
90	89	infeq1d 7140	. . . . . 6 |- ( z = b -> inf ( { z , R } , RR* , < ) = inf ( { b , R } , RR* , < ) )
91		simpr 110	. . . . . . 7 |- ( ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) -> b e. ( 0 [,] +oo ) )
92	91	adantl 277	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. ( 0 [,] +oo ) )
93		xrmincl 11692	. . . . . . 7 |- ( ( b e. RR* /\ R e. RR* ) -> inf ( { b , R } , RR* , < ) e. RR* )
94	78, 72, 93	syl2anc 411	. . . . . 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 5696	. . . . 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 4072	. . . 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 169	. . 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 275	. . . . . 6 |- ( a e. ( 0 [,] +oo ) -> 0 <_ a )
99	98	ad2antrl 490	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ a )
100	76	simprbi 275	. . . . . 6 |- ( b e. ( 0 [,] +oo ) -> 0 <_ b )
101	100	ad2antll 491	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ b )
102	41	adantr 276	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 < R )
103		xrbdtri 11702	. . . . 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 1266	. . . 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 3720	. . . . . 6 |- ( z = ( a +e b ) -> { z , R } = { ( a +e b ) , R } )
106	105	infeq1d 7140	. . . . 5 |- ( z = ( a +e b ) -> inf ( { z , R } , RR* , < ) = inf ( { ( a +e b ) , R } , RR* , < ) )
107		ge0xaddcl 10140	. . . . . 6 |- ( ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) -> ( a +e b ) e. ( 0 [,] +oo ) )
108	107	adantl 277	. . . . 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 10041	. . . . . 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 11692	. . . . . 6 |- ( ( ( a +e b ) e. RR* /\ R e. RR* ) -> inf ( { ( a +e b ) , R } , RR* , < ) e. RR* )
111	109, 72, 110	syl2anc 411	. . . . 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 5696	. . . 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 5985	. . . 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 4091	. . 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 15086	. 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 2285	1 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   {cpr 3644   class class class wbr 4059   |-> cmpt 4121   X. cxp 4691   o. ccom 4697   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   e. cmpo 5969  infcinf 7111   0cc0 7960   +oocpnf 8139   RR*cxr 8141   < clt 8142   <_ cle 8143   +ecxad 9927   [,]cicc 10048   *Metcxmet 14413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-map 6760  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-xneg 9929  df-xadd 9930  df-icc 10052  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-xmet 14421

This theorem is referenced by:  bdmet  15089  bdbl  15090  bdmopn  15091