Theorem txmetcnp 14258

Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)

Hypotheses

Ref	Expression
metcn.2	|- J = ( MetOpen ` C )
metcn.4	|- K = ( MetOpen ` D )
txmetcnp.4	|- L = ( MetOpen ` E )

Assertion

Ref	Expression
txmetcnp	|- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )

Distinct variable groups:   v, u, w, z, F   u, J, v, w, z   u, K, v, w, z   u, X, v, w, z   u, Y, v, w, z   u, Z, v, w, z   u, A, v, w, z   u, C, v, w, z   u, D, v, w, z   u, B, v, w, z   u, E, v, w, z   w, L, z

Allowed substitution hints:   L( v, u)

Proof of Theorem txmetcnp

Dummy variables t s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2187	. . . 4 |- ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) = ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) )
2		simp1 998	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> C e. ( *Met ` X ) )
3	2	adantr 276	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> C e. ( *Met ` X ) )
4		simp2 999	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> D e. ( *Met ` Y ) )
5	4	adantr 276	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> D e. ( *Met ` Y ) )
6	1, 3, 5	xmetxp 14247	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) e. ( *Met ` ( X X. Y ) ) )
7		simpl3 1003	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> E e. ( *Met ` Z ) )
8		simprl 529	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> A e. X )
9		simprr 531	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> B e. Y )
10	8, 9	opelxpd 4671	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> <. A , B >. e. ( X X. Y ) )
11		eqid 2187	. . . 4 |- ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) = ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) )
12		txmetcnp.4	. . . 4 |- L = ( MetOpen ` E )
13	11, 12	metcnp 14252	. . 3 |- ( ( ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) e. ( *Met ` ( X X. Y ) ) /\ E e. ( *Met ` Z ) /\ <. A , B >. e. ( X X. Y ) ) -> ( F e. ( ( ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) ) ) )
14	6, 7, 10, 13	syl3anc 1248	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) ) ) )
15		metcn.2	. . . . . 6 |- J = ( MetOpen ` C )
16		metcn.4	. . . . . 6 |- K = ( MetOpen ` D )
17	1, 3, 5, 15, 16, 11	xmettx 14250	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) = ( J tX K ) )
18	17	oveq1d 5903	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) CnP L ) = ( ( J tX K ) CnP L ) )
19	18	fveq1d 5529	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) CnP L ) ` <. A , B >. ) = ( ( ( J tX K ) CnP L ) ` <. A , B >. ) )
20	19	eleq2d 2257	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( MetOpen ` ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) ) CnP L ) ` <. A , B >. ) <-> F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) ) )
21		oveq2 5896	. . . . . . . . 9 |- ( t = <. u , v >. -> ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) = ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) )
22	21	breq1d 4025	. . . . . . . 8 |- ( t = <. u , v >. -> ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w <-> ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w ) )
23		fveq2 5527	. . . . . . . . . 10 |- ( t = <. u , v >. -> ( F ` t ) = ( F ` <. u , v >. ) )
24	23	oveq2d 5904	. . . . . . . . 9 |- ( t = <. u , v >. -> ( ( F ` <. A , B >. ) E ( F ` t ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) )
25	24	breq1d 4025	. . . . . . . 8 |- ( t = <. u , v >. -> ( ( ( F ` <. A , B >. ) E ( F ` t ) ) < z <-> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) )
26	22, 25	imbi12d 234	. . . . . . 7 |- ( t = <. u , v >. -> ( ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) <-> ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w -> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) ) )
27	26	ralxp 4782	. . . . . 6 |- ( A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) <-> A. u e. X A. v e. Y ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w -> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) )
28	8	ad4antr 494	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> A e. X )
29	9	ad4antr 494	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> B e. Y )
30	28, 29	opelxpd 4671	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> <. A , B >. e. ( X X. Y ) )
31		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> u e. X )
32		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> v e. Y )
33	31, 32	opelxpd 4671	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> <. u , v >. e. ( X X. Y ) )
34	2	ad5antr 496	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> C e. ( *Met ` X ) )
35		xmetf 14090	. . . . . . . . . . . . . . . 16 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
36	34, 35	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> C : ( X X. X ) --> RR* )
37		op1stg 6164	. . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y ) -> ( 1st ` <. A , B >. ) = A )
38	28, 29, 37	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 1st ` <. A , B >. ) = A )
39	38, 28	eqeltrd 2264	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 1st ` <. A , B >. ) e. X )
40		op1stg 6164	. . . . . . . . . . . . . . . . 17 |- ( ( u e. X /\ v e. Y ) -> ( 1st ` <. u , v >. ) = u )
41	40	adantll 476	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 1st ` <. u , v >. ) = u )
42	41, 31	eqeltrd 2264	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 1st ` <. u , v >. ) e. X )
43	36, 39, 42	fovcdmd 6032	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) e. RR* )
44	4	ad5antr 496	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> D e. ( *Met ` Y ) )
45		xmetf 14090	. . . . . . . . . . . . . . . 16 |- ( D e. ( *Met ` Y ) -> D : ( Y X. Y ) --> RR* )
46	44, 45	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> D : ( Y X. Y ) --> RR* )
47		op2ndg 6165	. . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y ) -> ( 2nd ` <. A , B >. ) = B )
48	28, 29, 47	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 2nd ` <. A , B >. ) = B )
49	48, 29	eqeltrd 2264	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 2nd ` <. A , B >. ) e. Y )
50		op2ndg 6165	. . . . . . . . . . . . . . . . 17 |- ( ( u e. X /\ v e. Y ) -> ( 2nd ` <. u , v >. ) = v )
51	50	adantll 476	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 2nd ` <. u , v >. ) = v )
52	51, 32	eqeltrd 2264	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( 2nd ` <. u , v >. ) e. Y )
53	46, 49, 52	fovcdmd 6032	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) e. RR* )
54		xrmaxcl 11273	. . . . . . . . . . . . . 14 |- ( ( ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) e. RR* /\ ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) e. RR* ) -> sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) e. RR* )
55	43, 53, 54	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) e. RR* )
56		fveq2 5527	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 1st ` r ) = ( 1st ` <. A , B >. ) )
57		fveq2 5527	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 1st ` s ) = ( 1st ` <. u , v >. ) )
58	56, 57	oveqan12d 5907	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 1st ` r ) C ( 1st ` s ) ) = ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) )
59		fveq2 5527	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 2nd ` r ) = ( 2nd ` <. A , B >. ) )
60		fveq2 5527	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 2nd ` s ) = ( 2nd ` <. u , v >. ) )
61	59, 60	oveqan12d 5907	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 2nd ` r ) D ( 2nd ` s ) ) = ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) )
62	58, 61	preq12d 3689	. . . . . . . . . . . . . . 15 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } = { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } )
63	62	supeq1d 6999	. . . . . . . . . . . . . 14 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) = sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) )
64	63, 1	ovmpoga 6017	. . . . . . . . . . . . 13 |- ( ( <. A , B >. e. ( X X. Y ) /\ <. u , v >. e. ( X X. Y ) /\ sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) e. RR* ) -> ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) = sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) )
65	30, 33, 55, 64	syl3anc 1248	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) = sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) )
66	38, 41	oveq12d 5906	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) = ( A C u ) )
67	48, 51	oveq12d 5906	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) = ( B D v ) )
68	66, 67	preq12d 3689	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } = { ( A C u ) , ( B D v ) } )
69	68	supeq1d 6999	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> sup ( { ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) , ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) } , RR* , < ) = sup ( { ( A C u ) , ( B D v ) } , RR* , < ) )
70	65, 69	eqtrd 2220	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) = sup ( { ( A C u ) , ( B D v ) } , RR* , < ) )
71	70	breq1d 4025	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w <-> sup ( { ( A C u ) , ( B D v ) } , RR* , < ) < w ) )
72		xmetcl 14092	. . . . . . . . . . . 12 |- ( ( C e. ( *Met ` X ) /\ A e. X /\ u e. X ) -> ( A C u ) e. RR* )
73	34, 28, 31, 72	syl3anc 1248	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( A C u ) e. RR* )
74		xmetcl 14092	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` Y ) /\ B e. Y /\ v e. Y ) -> ( B D v ) e. RR* )
75	44, 29, 32, 74	syl3anc 1248	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( B D v ) e. RR* )
76		rpxr 9674	. . . . . . . . . . . 12 |- ( w e. RR+ -> w e. RR* )
77	76	ad3antlr 493	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> w e. RR* )
78		xrmaxltsup 11279	. . . . . . . . . . 11 |- ( ( ( A C u ) e. RR* /\ ( B D v ) e. RR* /\ w e. RR* ) -> ( sup ( { ( A C u ) , ( B D v ) } , RR* , < ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
79	73, 75, 77, 78	syl3anc 1248	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( sup ( { ( A C u ) , ( B D v ) } , RR* , < ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
80	71, 79	bitrd 188	. . . . . . . . 9 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
81		df-ov 5891	. . . . . . . . . . . . 13 |- ( A F B ) = ( F ` <. A , B >. )
82		df-ov 5891	. . . . . . . . . . . . 13 |- ( u F v ) = ( F ` <. u , v >. )
83	81, 82	oveq12i 5900	. . . . . . . . . . . 12 |- ( ( A F B ) E ( u F v ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) )
84	83	breq1i 4022	. . . . . . . . . . 11 |- ( ( ( A F B ) E ( u F v ) ) < z <-> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z )
85	84	bicomi 132	. . . . . . . . . 10 |- ( ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z <-> ( ( A F B ) E ( u F v ) ) < z )
86	85	a1i 9	. . . . . . . . 9 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z <-> ( ( A F B ) E ( u F v ) ) < z ) )
87	80, 86	imbi12d 234	. . . . . . . 8 |- ( ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) /\ v e. Y ) -> ( ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w -> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) <-> ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
88	87	ralbidva 2483	. . . . . . 7 |- ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) /\ u e. X ) -> ( A. v e. Y ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w -> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) <-> A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
89	88	ralbidva 2483	. . . . . 6 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) -> ( A. u e. X A. v e. Y ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) <. u , v >. ) < w -> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) <-> A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
90	27, 89	bitrid 192	. . . . 5 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) /\ w e. RR+ ) -> ( A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) <-> A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
91	90	rexbidva 2484	. . . 4 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ z e. RR+ ) -> ( E. w e. RR+ A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) <-> E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
92	91	ralbidva 2483	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( A. z e. RR+ E. w e. RR+ A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) <-> A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
93	92	anbi2d 464	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. t e. ( X X. Y ) ( ( <. A , B >. ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) C ( 1st ` s ) ) , ( ( 2nd ` r ) D ( 2nd ` s ) ) } , RR* , < ) ) t ) < w -> ( ( F ` <. A , B >. ) E ( F ` t ) ) < z ) ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )
94	14, 20, 93	3bitr3d 218	1 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   A.wral 2465   E.wrex 2466   {cpr 3605   <.cop 3607   class class class wbr 4015   X. cxp 4636   -->wf 5224   ` cfv 5228  (class class class)co 5888   e. cmpo 5890   1stc1st 6152   2ndc2nd 6153   supcsup 6994   RR*cxr 8004   < clt 8005   RR+crp 9666   *Metcxmet 13666   MetOpencmopn 13671   CnP ccnp 13926   tX ctx 13992

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-map 6663  df-sup 6996  df-inf 6997  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-xneg 9785  df-xadd 9786  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-topgen 12726  df-psmet 13673  df-xmet 13674  df-bl 13676  df-mopn 13677  df-top 13738  df-topon 13751  df-bases 13783  df-cnp 13929  df-tx 13993

This theorem is referenced by:  txmetcn  14259  limccnp2cntop  14386