Theorem bdxmet 13295

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 992	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
2		xmetcl 13146	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR* )
3		xmetge0 13159	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x C y ) )
4		elxrge0 9935	. . . . . . 7 |- ( ( x C y ) e. ( 0 [,] +oo ) <-> ( ( x C y ) e. RR* /\ 0 <_ ( x C y ) ) )
5	2, 3, 4	sylanbrc 415	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. ( 0 [,] +oo ) )
6	5	3expb 1199	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
7	1, 6	sylan 281	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( x e. X /\ y e. X ) ) -> ( x C y ) e. ( 0 [,] +oo ) )
8		xmetf 13144	. . . . . . 7 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
9	8	3ad2ant1 1013	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
10	9	ffnd 5348	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C Fn ( X X. X ) )
11		fnovim 5961	. . . . 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 2171	. . . 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 3660	. . . . 5 |- ( z = ( x C y ) -> { z , R } = { ( x C y ) , R } )
15	14	infeq1d 6989	. . . 4 |- ( z = ( x C y ) -> inf ( { z , R } , RR* , < ) = inf ( { ( x C y ) , R } , RR* , < ) )
16	7, 12, 13, 15	fmpoco 6195	. . 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 2221	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) = D )
19		elxrge0 9935	. . . . . 6 |- ( z e. ( 0 [,] +oo ) <-> ( z e. RR* /\ 0 <_ z ) )
20	19	simplbi 272	. . . . 5 |- ( z e. ( 0 [,] +oo ) -> z e. RR* )
21		simp2 993	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> R e. RR* )
22		xrmincl 11229	. . . . 5 |- ( ( z e. RR* /\ R e. RR* ) -> inf ( { z , R } , RR* , < ) e. RR* )
23	20, 21, 22	syl2anr 288	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ z e. ( 0 [,] +oo ) ) -> inf ( { z , R } , RR* , < ) e. RR* )
24	23	fmpttd 5651	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) : ( 0 [,] +oo ) --> RR* )
25		eqid 2170	. . . . . 6 |- ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) = ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) )
26		preq1 3660	. . . . . . 7 |- ( z = a -> { z , R } = { a , R } )
27	26	infeq1d 6989	. . . . . 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 9935	. . . . . . . 8 |- ( a e. ( 0 [,] +oo ) <-> ( a e. RR* /\ 0 <_ a ) )
30	29	simplbi 272	. . . . . . 7 |- ( a e. ( 0 [,] +oo ) -> a e. RR* )
31		xrmincl 11229	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) e. RR* )
32	30, 21, 31	syl2anr 288	. . . . . 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 5589	. . . . 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 2179	. . . 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 7966	. . . . . . . . 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 275	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> a e. RR* )
38	21	adantr 274	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> R e. RR* )
39		xrltmininf 11233	. . . . . . . 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 1233	. . . . . . 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 994	. . . . . . . . 9 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> 0 < R )
42	41	adantr 274	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 < R )
43	42	biantrud 302	. . . . . . 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 662	. . . . 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 9755	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( 0 < R -> 0 <_ R ) )
49	35, 21, 48	sylancr 412	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ R )
52		xrlemininf 11234	. . . . . . . . . 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 1233	. . . . . . . . 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 939	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ inf ( { a , R } , RR* , < ) )
55		xrlenlt 7984	. . . . . . . . 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 412	. . . . . . . 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 303	. . . . . 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 9754	. . . . . . 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 409	. . . . . 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 7984	. . . . . . . . 9 |- ( ( 0 e. RR* /\ a e. RR* ) -> ( 0 <_ a <-> -. a < 0 ) )
63	35, 37, 62	sylancr 412	. . . . . . . 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 303	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> ( -. 0 < a <-> ( -. a < 0 /\ -. 0 < a ) ) )
66		xrlttri3 9754	. . . . . . 7 |- ( ( a e. RR* /\ 0 e. RR* ) -> ( a = 0 <-> ( -. a < 0 /\ -. 0 < a ) ) )
67	37, 36, 66	syl2anc 409	. . . . . 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 487	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> a e. RR* )
72	21	adantr 274	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> R e. RR* )
73		xrmin1inf 11230	. . . . . . . 8 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ a )
74	71, 72, 73	syl2anc 409	. . . . . . 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 409	. . . . . . . 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 9935	. . . . . . . . . 10 |- ( b e. ( 0 [,] +oo ) <-> ( b e. RR* /\ 0 <_ b ) )
77	76	simplbi 272	. . . . . . . . 9 |- ( b e. ( 0 [,] +oo ) -> b e. RR* )
78	77	ad2antll 488	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. RR* )
79		xrletr 9765	. . . . . . . 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 1233	. . . . . . 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 427	. . . . . 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 11231	. . . . . . 7 |- ( ( a e. RR* /\ R e. RR* ) -> inf ( { a , R } , RR* , < ) <_ R )
83	71, 72, 82	syl2anc 409	. . . . . 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 315	. . . . 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 11234	. . . . . 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 1233	. . . . 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 476	. . . . 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 3660	. . . . . . 7 |- ( z = b -> { z , R } = { b , R } )
90	89	infeq1d 6989	. . . . . 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 275	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. ( 0 [,] +oo ) )
93		xrmincl 11229	. . . . . . 7 |- ( ( b e. RR* /\ R e. RR* ) -> inf ( { b , R } , RR* , < ) e. RR* )
94	78, 72, 93	syl2anc 409	. . . . . 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 5589	. . . . 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 4002	. . . 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 273	. . . . . 6 |- ( a e. ( 0 [,] +oo ) -> 0 <_ a )
99	98	ad2antrl 487	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ a )
100	76	simprbi 273	. . . . . 6 |- ( b e. ( 0 [,] +oo ) -> 0 <_ b )
101	100	ad2antll 488	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 <_ b )
102	41	adantr 274	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> 0 < R )
103		xrbdtri 11239	. . . . 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 1249	. . . 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 3660	. . . . . 6 |- ( z = ( a +e b ) -> { z , R } = { ( a +e b ) , R } )
106	105	infeq1d 6989	. . . . 5 |- ( z = ( a +e b ) -> inf ( { z , R } , RR* , < ) = inf ( { ( a +e b ) , R } , RR* , < ) )
107		ge0xaddcl 9940	. . . . . 6 |- ( ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) -> ( a +e b ) e. ( 0 [,] +oo ) )
108	107	adantl 275	. . . . 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 9841	. . . . . 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 11229	. . . . . 6 |- ( ( ( a +e b ) e. RR* /\ R e. RR* ) -> inf ( { ( a +e b ) , R } , RR* , < ) e. RR* )
111	109, 72, 110	syl2anc 409	. . . . 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 5589	. . . 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 5871	. . . 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 4021	. . 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 13293	. 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 2248	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 973   = wceq 1348   e. wcel 2141   {cpr 3584   class class class wbr 3989   |-> cmpt 4050   X. cxp 4609   o. ccom 4615   Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853   e. cmpo 5855  infcinf 6960   0cc0 7774   +oocpnf 7951   RR*cxr 7953   < clt 7954   <_ cle 7955   +ecxad 9727   [,]cicc 9848   *Metcxmet 12774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-xneg 9729  df-xadd 9730  df-icc 9852  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-xmet 12782

This theorem is referenced by:  bdmet  13296  bdbl  13297  bdmopn  13298