Theorem xmetxpbl 12677

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 12676	. . 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 12558	. . 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 1216	. 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 6063	. . . . . . . . 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 6063	. . . . . . . . 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 12521	. . . . . . . 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 1216	. . . . . . 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 6064	. . . . . . . . 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 6064	. . . . . . . . 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 12521	. . . . . . . 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 1216	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
25		xrmaxcl 11021	. . . . . . 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 408	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) e. RR* )
27		fveq2 5421	. . . . . . . . . 10 |- ( u = C -> ( 1st ` u ) = ( 1st ` C ) )
28		fveq2 5421	. . . . . . . . . 10 |- ( v = t -> ( 1st ` v ) = ( 1st ` t ) )
29	27, 28	oveqan12d 5793	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` C ) M ( 1st ` t ) ) )
30		fveq2 5421	. . . . . . . . . 10 |- ( u = C -> ( 2nd ` u ) = ( 2nd ` C ) )
31		fveq2 5421	. . . . . . . . . 10 |- ( v = t -> ( 2nd ` v ) = ( 2nd ` t ) )
32	30, 31	oveqan12d 5793	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` C ) N ( 2nd ` t ) ) )
33	29, 32	preq12d 3608	. . . . . . . 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 6874	. . . . . . 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 5900	. . . . . 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 1216	. . . . 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 3939	. . . 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 11027	. . . . 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 1216	. . . 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 2674	. 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 6073	. . . . . . 7 |- ( n e. ( X X. Y ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
44	43	ad2antrl 481	. . . . . 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 6063	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 1st ` n ) e. X )
46	45	ad2antrl 481	. . . . . . 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 528	. . . . . . 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 12560	. . . . . . . . 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 1216	. . . . . . . 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 928	. . . . . 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 6064	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 2nd ` n ) e. Y )
54	53	ad2antrl 481	. . . . . . 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 529	. . . . . . 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 12560	. . . . . . . . 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 1216	. . . . . . . 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 928	. . . . . 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 520	. . . . . . . 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 528	. . . . . . . . . 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 529	. . . . . . . . . 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 6067	. . . . . . 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 585	. . . 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 5421	. . . . . . . 8 |- ( t = n -> ( 1st ` t ) = ( 1st ` n ) )
79	78	oveq2d 5790	. . . . . . 7 |- ( t = n -> ( ( 1st ` C ) M ( 1st ` t ) ) = ( ( 1st ` C ) M ( 1st ` n ) ) )
80	79	breq1d 3939	. . . . . 6 |- ( t = n -> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R <-> ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
81		fveq2 5421	. . . . . . . 8 |- ( t = n -> ( 2nd ` t ) = ( 2nd ` n ) )
82	81	oveq2d 5790	. . . . . . 7 |- ( t = n -> ( ( 2nd ` C ) N ( 2nd ` t ) ) = ( ( 2nd ` C ) N ( 2nd ` n ) ) )
83	82	breq1d 3939	. . . . . 6 |- ( t = n -> ( ( ( 2nd ` C ) N ( 2nd ` t ) ) < R <-> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) )
84	80, 83	anbi12d 464	. . . . 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 2840	. . . 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 6067	. . . 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 2137	. 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 2176	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 1331   e. wcel 1480   {crab 2420   {cpr 3528   <.cop 3530   class class class wbr 3929   X. cxp 4537   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   supcsup 6869   RR*cxr 7799   < clt 7800   *Metcxmet 12149   ballcbl 12151

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210

This theorem is referenced by:  xmettxlem  12678  xmettx  12679