Theorem bdxmet 13042

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 986	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
2		xmetcl 12893	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR* )
3		xmetge0 12906	. . . . . . 7 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x C y ) )
4		elxrge0 9905	. . . . . . 7 |- ( ( x C y ) e. ( 0 [,] +oo ) <-> ( ( x C y ) e. RR* /\ 0 <_ ( x C y ) ) )
5	2, 3, 4	sylanbrc 414	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. ( 0 [,] +oo ) )
6	5	3expb 1193	. . . . 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 12891	. . . . . . 7 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
9	8	3ad2ant1 1007	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
10	9	ffnd 5332	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C Fn ( X X. X ) )
11		fnovim 5941	. . . . 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 2165	. . . 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 3647	. . . . 5 |- ( z = ( x C y ) -> { z , R } = { ( x C y ) , R } )
15	14	infeq1d 6968	. . . 4 |- ( z = ( x C y ) -> inf ( { z , R } , RR* , < ) = inf ( { ( x C y ) , R } , RR* , < ) )
16	7, 12, 13, 15	fmpoco 6175	. . 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 2215	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) o. C ) = D )
19		elxrge0 9905	. . . . . 6 |- ( z e. ( 0 [,] +oo ) <-> ( z e. RR* /\ 0 <_ z ) )
20	19	simplbi 272	. . . . 5 |- ( z e. ( 0 [,] +oo ) -> z e. RR* )
21		simp2 987	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> R e. RR* )
22		xrmincl 11193	. . . . 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 5634	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) : ( 0 [,] +oo ) --> RR* )
25		eqid 2164	. . . . . 6 |- ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) ) = ( z e. ( 0 [,] +oo ) |-> inf ( { z , R } , RR* , < ) )
26		preq1 3647	. . . . . . 7 |- ( z = a -> { z , R } = { a , R } )
27	26	infeq1d 6968	. . . . . 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 9905	. . . . . . . 8 |- ( a e. ( 0 [,] +oo ) <-> ( a e. RR* /\ 0 <_ a ) )
30	29	simplbi 272	. . . . . . 7 |- ( a e. ( 0 [,] +oo ) -> a e. RR* )
31		xrmincl 11193	. . . . . . 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 5573	. . . . 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 2173	. . . 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 7936	. . . . . . . . 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 11197	. . . . . . . 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 1227	. . . . . . 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 988	. . . . . . . . 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 657	. . . . 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 9725	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( 0 < R -> 0 <_ R ) )
49	35, 21, 48	sylancr 411	. . . . . . . . . . 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 11198	. . . . . . . . . 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 1227	. . . . . . . . 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 933	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ a e. ( 0 [,] +oo ) ) -> 0 <_ inf ( { a , R } , RR* , < ) )
55		xrlenlt 7954	. . . . . . . . 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 411	. . . . . . . 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 9724	. . . . . . 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 7954	. . . . . . . . 9 |- ( ( 0 e. RR* /\ a e. RR* ) -> ( 0 <_ a <-> -. a < 0 ) )
63	35, 37, 62	sylancr 411	. . . . . . . 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 9724	. . . . . . 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 482	. . . . . . . 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 11194	. . . . . . . 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 9905	. . . . . . . . . 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 483	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( a e. ( 0 [,] +oo ) /\ b e. ( 0 [,] +oo ) ) ) -> b e. RR* )
79		xrletr 9735	. . . . . . . 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 1227	. . . . . . 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 426	. . . . . 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 11195	. . . . . . 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 11198	. . . . . 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 1227	. . . . 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 471	. . . . 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 3647	. . . . . . 7 |- ( z = b -> { z , R } = { b , R } )
90	89	infeq1d 6968	. . . . . 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 11193	. . . . . . 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 5573	. . . . 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 3989	. . . 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 482	. . . . 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 483	. . . . 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 11203	. . . . 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 1243	. . . 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 3647	. . . . . 6 |- ( z = ( a +e b ) -> { z , R } = { ( a +e b ) , R } )
106	105	infeq1d 6968	. . . . 5 |- ( z = ( a +e b ) -> inf ( { z , R } , RR* , < ) = inf ( { ( a +e b ) , R } , RR* , < ) )
107		ge0xaddcl 9910	. . . . . 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 9811	. . . . . 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 11193	. . . . . 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 5573	. . . 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 5854	. . . 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 4008	. . 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 13040	. 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 2242	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 967   = wceq 1342   e. wcel 2135   {cpr 3571   class class class wbr 3976   |-> cmpt 4037   X. cxp 4596   o. ccom 4602   Fn wfn 5177   -->wf 5178   ` cfv 5182  (class class class)co 5836   e. cmpo 5838  infcinf 6939   0cc0 7744   +oocpnf 7921   RR*cxr 7923   < clt 7924   <_ cle 7925   +ecxad 9697   [,]cicc 9818   *Metcxmet 12521

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-map 6607  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-rp 9581  df-xneg 9699  df-xadd 9700  df-icc 9822  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-xmet 12529

This theorem is referenced by:  bdmet  13043  bdbl  13044  bdmopn  13045