Theorem txmetcnp 13158

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 2165	. . . 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 987	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> C e. ( *Met ` X ) )
3	2	adantr 274	. . . 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 988	. . . . 5 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> D e. ( *Met ` Y ) )
5	4	adantr 274	. . . 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 13147	. . 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 992	. . 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 521	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> A e. X )
9		simprr 522	. . . 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 4637	. . 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 2165	. . . 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 13152	. . 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 1228	. 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 13150	. . . . 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 5857	. . . 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 5488	. . 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 2236	. 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 5850	. . . . . . . . 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 3992	. . . . . . . 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 5486	. . . . . . . . . 10 |- ( t = <. u , v >. -> ( F ` t ) = ( F ` <. u , v >. ) )
24	23	oveq2d 5858	. . . . . . . . 9 |- ( t = <. u , v >. -> ( ( F ` <. A , B >. ) E ( F ` t ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) )
25	24	breq1d 3992	. . . . . . . 8 |- ( t = <. u , v >. -> ( ( ( F ` <. A , B >. ) E ( F ` t ) ) < z <-> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z ) )
26	22, 25	imbi12d 233	. . . . . . 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 4747	. . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 4637	. . . . . . . . . . . . 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 520	. . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . 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 4637	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 12990	. . . . . . . . . . . . . . . 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 6118	. . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y ) -> ( 1st ` <. A , B >. ) = A )
38	28, 29, 37	syl2anc 409	. . . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . 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 6118	. . . . . . . . . . . . . . . . 17 |- ( ( u e. X /\ v e. Y ) -> ( 1st ` <. u , v >. ) = u )
41	40	adantll 468	. . . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . 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	fovrnd 5986	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 12990	. . . . . . . . . . . . . . . 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 6119	. . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y ) -> ( 2nd ` <. A , B >. ) = B )
48	28, 29, 47	syl2anc 409	. . . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . 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 6119	. . . . . . . . . . . . . . . . 17 |- ( ( u e. X /\ v e. Y ) -> ( 2nd ` <. u , v >. ) = v )
51	50	adantll 468	. . . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . 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	fovrnd 5986	. . . . . . . . . . . . . 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 11193	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 1st ` r ) = ( 1st ` <. A , B >. ) )
57		fveq2 5486	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 1st ` s ) = ( 1st ` <. u , v >. ) )
58	56, 57	oveqan12d 5861	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 1st ` r ) C ( 1st ` s ) ) = ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) )
59		fveq2 5486	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 2nd ` r ) = ( 2nd ` <. A , B >. ) )
60		fveq2 5486	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 2nd ` s ) = ( 2nd ` <. u , v >. ) )
61	59, 60	oveqan12d 5861	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 2nd ` r ) D ( 2nd ` s ) ) = ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) )
62	58, 61	preq12d 3661	. . . . . . . . . . . . . . 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 6952	. . . . . . . . . . . . . 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 5971	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . 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 5860	. . . . . . . . . . . . . 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 5860	. . . . . . . . . . . . . 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 3661	. . . . . . . . . . . . 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 6952	. . . . . . . . . . . 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 2198	. . . . . . . . . . 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 3992	. . . . . . . . . 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 12992	. . . . . . . . . . . 12 |- ( ( C e. ( *Met ` X ) /\ A e. X /\ u e. X ) -> ( A C u ) e. RR* )
73	34, 28, 31, 72	syl3anc 1228	. . . . . . . . . . 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 12992	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` Y ) /\ B e. Y /\ v e. Y ) -> ( B D v ) e. RR* )
75	44, 29, 32, 74	syl3anc 1228	. . . . . . . . . . 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 9597	. . . . . . . . . . . 12 |- ( w e. RR+ -> w e. RR* )
77	76	ad3antlr 485	. . . . . . . . . . 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 11199	. . . . . . . . . . 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 1228	. . . . . . . . . 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 187	. . . . . . . . 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 5845	. . . . . . . . . . . . 13 |- ( A F B ) = ( F ` <. A , B >. )
82		df-ov 5845	. . . . . . . . . . . . 13 |- ( u F v ) = ( F ` <. u , v >. )
83	81, 82	oveq12i 5854	. . . . . . . . . . . 12 |- ( ( A F B ) E ( u F v ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) )
84	83	breq1i 3989	. . . . . . . . . . 11 |- ( ( ( A F B ) E ( u F v ) ) < z <-> ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) < z )
85	84	bicomi 131	. . . . . . . . . 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 233	. . . . . . . 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 2462	. . . . . . 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 2462	. . . . . 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	syl5bb 191	. . . . 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 2463	. . . 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 2462	. . 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 460	. 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 217	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {cpr 3577   <.cop 3579   class class class wbr 3982   X. cxp 4602   -->wf 5184   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   supcsup 6947   RR*cxr 7932   < clt 7933   RR+crp 9589   *Metcxmet 12620   MetOpencmopn 12625   CnP ccnp 12826   tX ctx 12892

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-cnp 12829  df-tx 12893

This theorem is referenced by:  txmetcn  13159  limccnp2cntop  13286