Theorem xmetxpbl 15373

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 15372	. . 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 15254	. . 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 1274	. 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 6359	. . . . . . . . 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 6359	. . . . . . . . 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 15217	. . . . . . . 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 1274	. . . . . . 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 6360	. . . . . . . . 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 6360	. . . . . . . . 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 15217	. . . . . . . 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 1274	. . . . . . 7 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( ( 2nd ` C ) N ( 2nd ` t ) ) e. RR* )
25		xrmaxcl 11937	. . . . . . 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 5670	. . . . . . . . . 10 |- ( u = C -> ( 1st ` u ) = ( 1st ` C ) )
28		fveq2 5670	. . . . . . . . . 10 |- ( v = t -> ( 1st ` v ) = ( 1st ` t ) )
29	27, 28	oveqan12d 6069	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` C ) M ( 1st ` t ) ) )
30		fveq2 5670	. . . . . . . . . 10 |- ( u = C -> ( 2nd ` u ) = ( 2nd ` C ) )
31		fveq2 5670	. . . . . . . . . 10 |- ( v = t -> ( 2nd ` v ) = ( 2nd ` t ) )
32	30, 31	oveqan12d 6069	. . . . . . . . 9 |- ( ( u = C /\ v = t ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` C ) N ( 2nd ` t ) ) )
33	29, 32	preq12d 3776	. . . . . . . 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 7278	. . . . . . 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 6183	. . . . . 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 1274	. . . . 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 4119	. . . 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 11943	. . . . 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 1274	. . . 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 2801	. 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 6369	. . . . . . 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 6359	. . . . . . . 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 541	. . . . . . 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 15256	. . . . . . . . 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 1274	. . . . . . . 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 953	. . . . . 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 6360	. . . . . . . 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 542	. . . . . . 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 15256	. . . . . . . . 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 1274	. . . . . . . 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 953	. . . . . 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 531	. . . . . . . 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 541	. . . . . . . . . 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 542	. . . . . . . . . 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 6363	. . . . . . 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 600	. . . 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 5670	. . . . . . . 8 |- ( t = n -> ( 1st ` t ) = ( 1st ` n ) )
79	78	oveq2d 6066	. . . . . . 7 |- ( t = n -> ( ( 1st ` C ) M ( 1st ` t ) ) = ( ( 1st ` C ) M ( 1st ` n ) ) )
80	79	breq1d 4119	. . . . . 6 |- ( t = n -> ( ( ( 1st ` C ) M ( 1st ` t ) ) < R <-> ( ( 1st ` C ) M ( 1st ` n ) ) < R ) )
81		fveq2 5670	. . . . . . . 8 |- ( t = n -> ( 2nd ` t ) = ( 2nd ` n ) )
82	81	oveq2d 6066	. . . . . . 7 |- ( t = n -> ( ( 2nd ` C ) N ( 2nd ` t ) ) = ( ( 2nd ` C ) N ( 2nd ` n ) ) )
83	82	breq1d 4119	. . . . . 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 2973	. . . 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 6363	. . . 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 2230	. 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 2269	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 1398   e. wcel 2203   {crab 2524   {cpr 3690   <.cop 3692   class class class wbr 4109   X. cxp 4747   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   1stc1st 6332   2ndc2nd 6333   supcsup 7273   RR*cxr 8307   < clt 8308   *Metcxmet 14684   ballcbl 14686

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-xneg 10105  df-xadd 10106  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-topgen 13473  df-psmet 14691  df-xmet 14692  df-bl 14694  df-mopn 14695  df-top 14863  df-topon 14876  df-bases 14908

This theorem is referenced by:  xmettxlem  15374  xmettx  15375