Theorem xmetxpbl 14011

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 14010	. . 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 13892	. . 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 1238	. 2 |- ( ph -> ( C ( ball ` P ) R ) = { t e. ( X X. Y ) | ( C P t ) < R } )
9	5	adantr 276	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> C e. ( X X. Y ) )
10		simpr 110	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> t e. ( X X. Y ) )
11	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> M e. ( *Met ` X ) )
12		xp1st 6166	. . . . . . . . 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 6166	. . . . . . . . 9 |- ( t e. ( X X. Y ) -> ( 1st ` t ) e. X )
15	14	adantl 277	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 1st ` t ) e. X )
16		xmetcl 13855	. . . . . . . 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 1238	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 1st ` C ) M ( 1st ` t ) ) e. RR* )
18	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> N e. ( *Met ` Y ) )
19		xp2nd 6167	. . . . . . . . 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 6167	. . . . . . . . 9 |- ( t e. ( X X. Y ) -> ( 2nd ` t ) e. Y )
22	21	adantl 277	. . . . . . . 8 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 2nd ` t ) e. Y )
23		xmetcl 13855	. . . . . . . 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 1238	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
25		xrmaxcl 11260	. . . . . . 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 411	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> sup ( { ( ( 1st ` C ) M ( 1st ` t ) ) , ( ( 2nd ` C ) N ( 2nd ` t ) ) } , RR* , < ) e. RR* )
27		fveq2 5516	. . . . . . . . . 10 |- ( u = C -> ( 1st ` u ) = ( 1st ` C ) )
28		fveq2 5516	. . . . . . . . . 10 |- ( v = t -> ( 1st ` v ) = ( 1st ` t ) )
29	27, 28	oveqan12d 5894	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` C ) M ( 1st ` t ) ) )
30		fveq2 5516	. . . . . . . . . 10 |- ( u = C -> ( 2nd ` u ) = ( 2nd ` C ) )
31		fveq2 5516	. . . . . . . . . 10 |- ( v = t -> ( 2nd ` v ) = ( 2nd ` t ) )
32	30, 31	oveqan12d 5894	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` C ) N ( 2nd ` t ) ) )
33	29, 32	preq12d 3678	. . . . . . . 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 6986	. . . . . . 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 6004	. . . . . 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 1238	. . . . 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 4014	. . . 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 276	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> R e. RR* )
39		xrmaxltsup 11266	. . . . 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 1238	. . . 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 188	. . 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 2726	. 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 6176	. . . . . . 7 |- ( n e. ( X X. Y ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
44	43	ad2antrl 490	. . . . . 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 6166	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 1st ` n ) e. X )
46	45	ad2antrl 490	. . . . . . 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 539	. . . . . . 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 13894	. . . . . . . . 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 1238	. . . . . . . 8 |- ( ph -> ( ( 1st ` n ) e. ( ( 1st ` C ) ( ball ` M ) R ) <-> ( ( 1st ` n ) e. X /\ ( ( 1st ` C ) M ( 1st ` n ) ) < R ) ) )
51	50	adantr 276	. . . . . . 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 944	. . . . . 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 6167	. . . . . . . 8 |- ( n e. ( X X. Y ) -> ( 2nd ` n ) e. Y )
54	53	ad2antrl 490	. . . . . . 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 540	. . . . . . 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 13894	. . . . . . . . 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 1238	. . . . . . . 8 |- ( ph -> ( ( 2nd ` n ) e. ( ( 2nd ` C ) ( ball ` N ) R ) <-> ( ( 2nd ` n ) e. Y /\ ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) ) )
59	58	adantr 276	. . . . . . 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 944	. . . . . 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 310	. . . . 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 529	. . . . . . . 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 539	. . . . . . . . . 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 276	. . . . . . . . . 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 147	. . . . . . . . 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 112	. . . . . . . 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 540	. . . . . . . . . 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 276	. . . . . . . . . 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 147	. . . . . . . . 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 112	. . . . . . . 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 310	. . . . . . 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 6170	. . . . . . 7 |- ( n e. ( X X. Y ) <-> ( n = <. ( 1st ` n ) , ( 2nd ` n ) >. /\ ( ( 1st ` n ) e. X /\ ( 2nd ` n ) e. Y ) ) )
73	71, 72	sylibr 134	. . . . . 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 114	. . . . . 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 114	. . . . . 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 310	. . . . 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 596	. . . 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 5516	. . . . . . . 8 |- ( t = n -> ( 1st ` t ) = ( 1st ` n ) )
79	78	oveq2d 5891	. . . . . . 7 |- ( t = n -> ( ( 1st ` C ) M ( 1st ` t ) ) = ( ( 1st ` C ) M ( 1st ` n ) ) )
80	79	breq1d 4014	. . . . . 6 |- ( t = n -> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R <-> ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
81		fveq2 5516	. . . . . . . 8 |- ( t = n -> ( 2nd ` t ) = ( 2nd ` n ) )
82	81	oveq2d 5891	. . . . . . 7 |- ( t = n -> ( ( 2nd ` C ) N ( 2nd ` t ) ) = ( ( 2nd ` C ) N ( 2nd ` n ) ) )
83	82	breq1d 4014	. . . . . 6 |- ( t = n -> ( ( ( 2nd ` C ) N ( 2nd ` t ) ) < R <-> ( ( 2nd ` C ) N ( 2nd ` n ) ) < R ) )
84	80, 83	anbi12d 473	. . . . 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 2894	. . . 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 6170	. . . 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 223	. . 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 2175	. 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 2214	1 |- ( ph -> ( C ( ball ` P ) R ) = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {crab 2459   {cpr 3594   <.cop 3596   class class class wbr 4004   X. cxp 4625   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   1stc1st 6139   2ndc2nd 6140   supcsup 6981   RR*cxr 7991   < clt 7992   *Metcxmet 13443   ballcbl 13445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-map 6650  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-xneg 9772  df-xadd 9773  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-topgen 12709  df-psmet 13450  df-xmet 13451  df-bl 13453  df-mopn 13454  df-top 13501  df-topon 13514  df-bases 13546

This theorem is referenced by:  xmettxlem  14012  xmettx  14013