Theorem xmetxpbl 12868

Description: The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point C with radius R. (Contributed by Jim Kingdon, 22-Oct-2023.)

Hypotheses

Ref	Expression
xmetxp.p	|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
xmetxp.1	|- ( ph -> M e. ( *Met ` X ) )
xmetxp.2	|- ( ph -> N e. ( *Met ` Y ) )
xmetxpbl.r	|- ( ph -> R e. RR* )
xmetxpbl.c	|- ( ph -> C e. ( X X. Y ) )

Assertion

Ref	Expression
xmetxpbl	|- ( ph -> ( C ( ball ` P ) R ) = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )

Distinct variable groups:   u, C, v   u, M, v   u, N, v   u, X, v   u, Y, v

Allowed substitution hints:   ph( v, u)   P( v, u)   R( v, u)

Proof of Theorem xmetxpbl

Dummy variables n t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.p	. . . 4 |- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
2		xmetxp.1	. . . 4 |- ( ph -> M e. ( *Met ` X ) )
3		xmetxp.2	. . . 4 |- ( ph -> N e. ( *Met ` Y ) )
4	1, 2, 3	xmetxp 12867	. . 3 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		xmetxpbl.c	. . 3 |- ( ph -> C e. ( X X. Y ) )
6		xmetxpbl.r	. . 3 |- ( ph -> R e. RR* )
7		blval 12749	. . 3 |- ( ( P e. ( *Met ` ( X X. Y ) ) /\ C e. ( X X. Y ) /\ R e. RR* ) -> ( C ( ball ` P ) R ) = { t e. ( X X. Y ) | ( C P t ) < R } )
8	4, 5, 6, 7	syl3anc 1220	. 2 |- ( ph -> ( C ( ball ` P ) R ) = { t e. ( X X. Y ) | ( C P t ) < R } )
9	5	adantr 274	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> C e. ( X X. Y ) )
10		simpr 109	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> t e. ( X X. Y ) )
11	2	adantr 274	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> M e. ( *Met ` X ) )
12		xp1st 6107	. . . . . . . . 9 |- ( C e. ( X X. Y ) -> ( 1st ` C ) e. X )
13	9, 12	syl 14	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 1st ` C ) e. X )
14		xp1st 6107	. . . . . . . . 9 |- ( t e. ( X X. Y ) -> ( 1st ` t ) e. X )
15	14	adantl 275	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 1st ` t ) e. X )
16		xmetcl 12712	. . . . . . . 8 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` C ) e. X /\ ( 1st ` t ) e. X ) -> ( ( 1st ` C ) M ( 1st ` t ) ) e. RR* )
17	11, 13, 15, 16	syl3anc 1220	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 1st ` C ) M ( 1st ` t ) ) e. RR* )
18	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> N e. ( *Met ` Y ) )
19		xp2nd 6108	. . . . . . . . 9 |- ( C e. ( X X. Y ) -> ( 2nd ` C ) e. Y )
20	9, 19	syl 14	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 2nd ` C ) e. Y )
21		xp2nd 6108	. . . . . . . . 9 |- ( t e. ( X X. Y ) -> ( 2nd ` t ) e. Y )
22	21	adantl 275	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 2nd ` t ) e. Y )
23		xmetcl 12712	. . . . . . . 8 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` C ) e. Y /\ ( 2nd ` t ) e. Y ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
24	18, 20, 22, 23	syl3anc 1220	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
25		xrmaxcl 11131	. . . . . . 7 |- ( ( ( ( 1st ` C ) M ( 1st ` t ) ) e. RR* /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* ) -> sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) e. RR* )
26	17, 24, 25	syl2anc 409	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) e. RR* )
27		fveq2 5465	. . . . . . . . . 10 |- ( u = C -> ( 1st ` u ) = ( 1st ` C ) )
28		fveq2 5465	. . . . . . . . . 10 |- ( v = t -> ( 1st ` v ) = ( 1st ` t ) )
29	27, 28	oveqan12d 5837	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` C ) M ( 1st ` t ) ) )
30		fveq2 5465	. . . . . . . . . 10 |- ( u = C -> ( 2nd ` u ) = ( 2nd ` C ) )
31		fveq2 5465	. . . . . . . . . 10 |- ( v = t -> ( 2nd ` v ) = ( 2nd ` t ) )
32	30, 31	oveqan12d 5837	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` C ) N ( 2nd ` t ) ) )
33	29, 32	preq12d 3644	. . . . . . . 8 |- ( ( u = C /\ v = t ) -> { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } = { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } )
34	33	supeq1d 6923	. . . . . . 7 |- ( ( u = C /\ v = t ) -> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) = sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) )
35	34, 1	ovmpoga 5944	. . . . . 6 |- ( ( C e. ( X X. Y ) /\ t e. ( X X. Y ) /\ sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) e. RR* ) -> ( C P t ) = sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) )
36	9, 10, 26, 35	syl3anc 1220	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( C P t ) = sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) )
37	36	breq1d 3975	. . . 4 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( C P t ) < R <-> sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) < R ) )
38	6	adantr 274	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> R e. RR* )
39		xrmaxltsup 11137	. . . . 5 |- ( ( ( ( 1st ` C ) M ( 1st ` t ) ) e. RR* /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* /\ R e. RR* ) -> ( sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) < R <-> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) ) )
40	17, 24, 38, 39	syl3anc 1220	. . . 4 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) < R <-> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) ) )
41	37, 40	bitrd 187	. . 3 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( C P t ) < R <-> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) ) )
42	41	rabbidva 2700	. 2 |- ( ph -> { t e. ( X X. Y ) | ( C P t ) < R } = { t e. ( X X. Y ) | ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) } )
43		1st2nd2 6117	. . . . . . 7 |- ( n e. ( X X. Y ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
44	43	ad2antrl 482	. . . . . 6 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
45		xp1st 6107	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 1st ` n ) e. X )
46	45	ad2antrl 482	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( 1st ` n ) e. X )
47		simprrl 529	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( ( 1st ` C ) M ( 1st ` n ) ) < R )
48	5, 12	syl 14	. . . . . . . . 9 |- ( ph -> ( 1st ` C ) e. X )
49		elbl 12751	. . . . . . . . 9 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` C ) e. X /\ R e. RR* ) -> ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) <-> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) ) )
50	2, 48, 6, 49	syl3anc 1220	. . . . . . . 8 |- ( ph -> ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) <-> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) ) )
51	50	adantr 274	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) <-> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) ) )
52	46, 47, 51	mpbir2and 929	. . . . . 6 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) )
53		xp2nd 6108	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 2nd ` n ) e. Y )
54	53	ad2antrl 482	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( 2nd ` n ) e. Y )
55		simprrr 530	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R )
56	5, 19	syl 14	. . . . . . . . 9 |- ( ph -> ( 2nd ` C ) e. Y )
57		elbl 12751	. . . . . . . . 9 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` C ) e. Y /\ R e. RR* ) -> ( ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) <-> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
58	3, 56, 6, 57	syl3anc 1220	. . . . . . . 8 |- ( ph -> ( ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) <-> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
59	58	adantr 274	. . . . . . 7 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) <-> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
60	54, 55, 59	mpbir2and 929	. . . . . 6 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) )
61	44, 52, 60	jca32 308	. . . . 5 |- ( ( ph /\ ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) ) -> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) )
62		simprl 521	. . . . . . . 8 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
63		simprrl 529	. . . . . . . . . 10 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) )
64	50	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) <-> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) ) )
65	63, 64	mpbid 146	. . . . . . . . 9 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
66	65	simpld 111	. . . . . . . 8 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( 1st ` n ) e. X )
67		simprrr 530	. . . . . . . . . 10 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) )
68	58	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) <-> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
69	67, 68	mpbid 146	. . . . . . . . 9 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) )
70	69	simpld 111	. . . . . . . 8 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( 2nd ` n ) e. Y )
71	62, 66, 70	jca32 308	. . . . . . 7 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. X /\ ( 2nd ` n ) e. Y ) ) )
72		elxp6 6111	. . . . . . 7 |- ( n e. ( X X. Y ) <-> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. X /\ ( 2nd ` n ) e. Y ) ) )
73	71, 72	sylibr 133	. . . . . 6 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> n e. ( X X. Y ) )
74	65	simprd 113	. . . . . 6 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 1st ` C ) M ( 1st ` n ) ) < R )
75	69	simprd 113	. . . . . 6 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R )
76	73, 74, 75	jca32 308	. . . . 5 |- ( ( ph /\ ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) -> ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
77	61, 76	impbida 586	. . . 4 |- ( ph -> ( ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) <-> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) ) )
78		fveq2 5465	. . . . . . . 8 |- ( t = n -> ( 1st ` t ) = ( 1st ` n ) )
79	78	oveq2d 5834	. . . . . . 7 |- ( t = n -> ( ( 1st ` C ) M ( 1st ` t ) ) = ( ( 1st ` C ) M ( 1st ` n ) ) )
80	79	breq1d 3975	. . . . . 6 |- ( t = n -> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R <-> ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
81		fveq2 5465	. . . . . . . 8 |- ( t = n -> ( 2nd ` t ) = ( 2nd ` n ) )
82	81	oveq2d 5834	. . . . . . 7 |- ( t = n -> ( ( 2nd ` C ) N ( 2nd ` t ) ) = ( ( 2nd ` C ) N ( 2nd ` n ) ) )
83	82	breq1d 3975	. . . . . 6 |- ( t = n -> ( ( ( 2nd ` C ) N ( 2nd ` t ) ) < R <-> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) )
84	80, 83	anbi12d 465	. . . . 5 |- ( t = n -> ( ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) <-> ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
85	84	elrab 2868	. . . 4 |- ( n e. { t e. ( X X. Y ) | ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) } <-> ( n e. ( X X. Y ) /\ ( ( ( 1st ` C ) M ( 1st ` n ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
86		elxp6 6111	. . . 4 |- ( n e. ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) <-> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) /\ ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) )
87	77, 85, 86	3bitr4g 222	. . 3 |- ( ph -> ( n e. { t e. ( X X. Y ) | ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) } <-> n e. ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) ) )
88	87	eqrdv 2155	. 2 |- ( ph -> { t e. ( X X. Y ) | ( ( ( 1st ` C ) M ( 1st ` t ) ) < R /\ ( ( 2nd ` C ) N ( 2nd ` t ) ) < R ) } = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )
89	8, 42, 88	3eqtrd 2194	1 |- ( ph -> ( C ( ball ` P ) R ) = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {crab 2439   {cpr 3561   <.cop 3563   class class class wbr 3965   X. cxp 4581   ` cfv 5167  (class class class)co 5818   e. cmpo 5820   1stc1st 6080   2ndc2nd 6081   supcsup 6918   RR*cxr 7894   < clt 7895   *Metcxmet 12340   ballcbl 12342

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-map 6588  df-sup 6920  df-inf 6921  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-xneg 9661  df-xadd 9662  df-seqfrec 10327  df-exp 10401  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-topgen 12332  df-psmet 12347  df-xmet 12348  df-bl 12350  df-mopn 12351  df-top 12356  df-topon 12369  df-bases 12401

This theorem is referenced by:  xmettxlem  12869  xmettx  12870