Theorem txmetcnp 14754

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 2196	. . . 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 999	. . . . 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 1000	. . . . 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 14743	. . 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 1004	. . 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 4696	. . 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 2196	. . . 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 14748	. . 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 1249	. 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 14746	. . . . 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 5937	. . . 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 5560	. . 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 2266	. 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 5930	. . . . . . . . 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 4043	. . . . . . . 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 5558	. . . . . . . . . 10 |- ( t = <. u , v >. -> ( F ` t ) = ( F ` <. u , v >. ) )
24	23	oveq2d 5938	. . . . . . . . 9 |- ( t = <. u , v >. -> ( ( F ` <. A , B >. ) E ( F ` t ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) )
25	24	breq1d 4043	. . . . . . . 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 4809	. . . . . 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 4696	. . . . . . . . . . . . 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 4696	. . . . . . . . . . . . 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 14586	. . . . . . . . . . . . . . . 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 6208	. . . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . . 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 6208	. . . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . . 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 6068	. . . . . . . . . . . . . 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 14586	. . . . . . . . . . . . . . . 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 6209	. . . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . . 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 6209	. . . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . . 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 6068	. . . . . . . . . . . . . 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 11417	. . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 1st ` r ) = ( 1st ` <. A , B >. ) )
57		fveq2 5558	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 1st ` s ) = ( 1st ` <. u , v >. ) )
58	56, 57	oveqan12d 5941	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 1st ` r ) C ( 1st ` s ) ) = ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) )
59		fveq2 5558	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 2nd ` r ) = ( 2nd ` <. A , B >. ) )
60		fveq2 5558	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 2nd ` s ) = ( 2nd ` <. u , v >. ) )
61	59, 60	oveqan12d 5941	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 2nd ` r ) D ( 2nd ` s ) ) = ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) )
62	58, 61	preq12d 3707	. . . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . . 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 6052	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 5940	. . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . 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 3707	. . . . . . . . . . . . 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 7053	. . . . . . . . . . . 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 2229	. . . . . . . . . . 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 4043	. . . . . . . . . 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 14588	. . . . . . . . . . . 12 |- ( ( C e. ( *Met ` X ) /\ A e. X /\ u e. X ) -> ( A C u ) e. RR* )
73	34, 28, 31, 72	syl3anc 1249	. . . . . . . . . . 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 14588	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` Y ) /\ B e. Y /\ v e. Y ) -> ( B D v ) e. RR* )
75	44, 29, 32, 74	syl3anc 1249	. . . . . . . . . . 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 9736	. . . . . . . . . . . 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 11423	. . . . . . . . . . 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 1249	. . . . . . . . . 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 5925	. . . . . . . . . . . . 13 |- ( A F B ) = ( F ` <. A , B >. )
82		df-ov 5925	. . . . . . . . . . . . 13 |- ( u F v ) = ( F ` <. u , v >. )
83	81, 82	oveq12i 5934	. . . . . . . . . . . 12 |- ( ( A F B ) E ( u F v ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) )
84	83	breq1i 4040	. . . . . . . . . . 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 2493	. . . . . . 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 2493	. . . . . 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 2494	. . . 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 2493	. . 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {cpr 3623   <.cop 3625   class class class wbr 4033   X. cxp 4661   -->wf 5254   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197   supcsup 7048   RR*cxr 8060   < clt 8061   RR+crp 9728   *Metcxmet 14092   MetOpencmopn 14097   CnP ccnp 14422   tX ctx 14488

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-cnp 14425  df-tx 14489

This theorem is referenced by:  txmetcn  14755  limccnp2cntop  14913