Theorem xmetxpbl 13302

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 13301	. . 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 13183	. . 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 1233	. 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 6144	. . . . . . . . 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 6144	. . . . . . . . 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 13146	. . . . . . . 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 1233	. . . . . . 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 6145	. . . . . . . . 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 6145	. . . . . . . . 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 13146	. . . . . . . 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 1233	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
25		xrmaxcl 11215	. . . . . . 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 5496	. . . . . . . . . 10 |- ( u = C -> ( 1st ` u ) = ( 1st ` C ) )
28		fveq2 5496	. . . . . . . . . 10 |- ( v = t -> ( 1st ` v ) = ( 1st ` t ) )
29	27, 28	oveqan12d 5872	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` C ) M ( 1st ` t ) ) )
30		fveq2 5496	. . . . . . . . . 10 |- ( u = C -> ( 2nd ` u ) = ( 2nd ` C ) )
31		fveq2 5496	. . . . . . . . . 10 |- ( v = t -> ( 2nd ` v ) = ( 2nd ` t ) )
32	30, 31	oveqan12d 5872	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` C ) N ( 2nd ` t ) ) )
33	29, 32	preq12d 3668	. . . . . . . 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 6964	. . . . . . 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 5982	. . . . . 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 1233	. . . . 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 3999	. . . 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 11221	. . . . 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 1233	. . . 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 2718	. 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 6154	. . . . . . 7 |- ( n e. ( X X. Y ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
44	43	ad2antrl 487	. . . . . 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 6144	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 1st ` n ) e. X )
46	45	ad2antrl 487	. . . . . . 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 534	. . . . . . 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 13185	. . . . . . . . 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 1233	. . . . . . . 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 939	. . . . . 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 6145	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 2nd ` n ) e. Y )
54	53	ad2antrl 487	. . . . . . 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 535	. . . . . . 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 13185	. . . . . . . . 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 1233	. . . . . . . 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 939	. . . . . 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 526	. . . . . . . 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 534	. . . . . . . . . 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 535	. . . . . . . . . 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 6148	. . . . . . 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 591	. . . 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 5496	. . . . . . . 8 |- ( t = n -> ( 1st ` t ) = ( 1st ` n ) )
79	78	oveq2d 5869	. . . . . . 7 |- ( t = n -> ( ( 1st ` C ) M ( 1st ` t ) ) = ( ( 1st ` C ) M ( 1st ` n ) ) )
80	79	breq1d 3999	. . . . . 6 |- ( t = n -> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R <-> ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
81		fveq2 5496	. . . . . . . 8 |- ( t = n -> ( 2nd ` t ) = ( 2nd ` n ) )
82	81	oveq2d 5869	. . . . . . 7 |- ( t = n -> ( ( 2nd ` C ) N ( 2nd ` t ) ) = ( ( 2nd ` C ) N ( 2nd ` n ) ) )
83	82	breq1d 3999	. . . . . 6 |- ( t = n -> ( ( ( 2nd ` C ) N ( 2nd ` t ) ) < R <-> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) )
84	80, 83	anbi12d 470	. . . . 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 2886	. . . 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 6148	. . . 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 2168	. 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 2207	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 1348   e. wcel 2141   {crab 2452   {cpr 3584   <.cop 3586   class class class wbr 3989   X. cxp 4609   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   supcsup 6959   RR*cxr 7953   < clt 7954   *Metcxmet 12774   ballcbl 12776

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-stab 826  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-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835

This theorem is referenced by:  xmettxlem  13303  xmettx  13304