Theorem txmetcnp 12865

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 2154	. . . 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 982	. . . . 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 983	. . . . 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 12854	. . 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 987	. . 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 4612	. . 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 2154	. . . 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 12859	. . 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 1217	. 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 12857	. . . . 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 5829	. . . 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 5463	. . 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 2224	. 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 5822	. . . . . . . . 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 3971	. . . . . . . 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 5461	. . . . . . . . . 10 |- ( t = <. u , v >. -> ( F ` t ) = ( F ` <. u , v >. ) )
24	23	oveq2d 5830	. . . . . . . . 9 |- ( t = <. u , v >. -> ( ( F ` <. A , B >. ) E ( F ` t ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) ) )
25	24	breq1d 3971	. . . . . . . 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 4722	. . . . . 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 4612	. . . . . . . . . . . . 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 4612	. . . . . . . . . . . . 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 12697	. . . . . . . . . . . . . . . 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 6088	. . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 6088	. . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 5955	. . . . . . . . . . . . . 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 12697	. . . . . . . . . . . . . . . 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 6089	. . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 6089	. . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 5955	. . . . . . . . . . . . . 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 11126	. . . . . . . . . . . . . 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 5461	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 1st ` r ) = ( 1st ` <. A , B >. ) )
57		fveq2 5461	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 1st ` s ) = ( 1st ` <. u , v >. ) )
58	56, 57	oveqan12d 5833	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 1st ` r ) C ( 1st ` s ) ) = ( ( 1st ` <. A , B >. ) C ( 1st ` <. u , v >. ) ) )
59		fveq2 5461	. . . . . . . . . . . . . . . . 17 |- ( r = <. A , B >. -> ( 2nd ` r ) = ( 2nd ` <. A , B >. ) )
60		fveq2 5461	. . . . . . . . . . . . . . . . 17 |- ( s = <. u , v >. -> ( 2nd ` s ) = ( 2nd ` <. u , v >. ) )
61	59, 60	oveqan12d 5833	. . . . . . . . . . . . . . . 16 |- ( ( r = <. A , B >. /\ s = <. u , v >. ) -> ( ( 2nd ` r ) D ( 2nd ` s ) ) = ( ( 2nd ` <. A , B >. ) D ( 2nd ` <. u , v >. ) ) )
62	58, 61	preq12d 3640	. . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . 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 5832	. . . . . . . . . . . . . 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 5832	. . . . . . . . . . . . . 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 3640	. . . . . . . . . . . . 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 6919	. . . . . . . . . . . 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 2187	. . . . . . . . . . 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 3971	. . . . . . . . . 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 12699	. . . . . . . . . . . 12 |- ( ( C e. ( *Met ` X ) /\ A e. X /\ u e. X ) -> ( A C u ) e. RR* )
73	34, 28, 31, 72	syl3anc 1217	. . . . . . . . . . 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 12699	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` Y ) /\ B e. Y /\ v e. Y ) -> ( B D v ) e. RR* )
75	44, 29, 32, 74	syl3anc 1217	. . . . . . . . . . 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 9546	. . . . . . . . . . . 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 11132	. . . . . . . . . . 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 1217	. . . . . . . . . 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 5817	. . . . . . . . . . . . 13 |- ( A F B ) = ( F ` <. A , B >. )
82		df-ov 5817	. . . . . . . . . . . . 13 |- ( u F v ) = ( F ` <. u , v >. )
83	81, 82	oveq12i 5826	. . . . . . . . . . . 12 |- ( ( A F B ) E ( u F v ) ) = ( ( F ` <. A , B >. ) E ( F ` <. u , v >. ) )
84	83	breq1i 3968	. . . . . . . . . . 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 2450	. . . . . . 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 2450	. . . . . 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 2451	. . . 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 2450	. . 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 963   = wceq 1332   e. wcel 2125   A.wral 2432   E.wrex 2433   {cpr 3557   <.cop 3559   class class class wbr 3961   X. cxp 4577   -->wf 5159   ` cfv 5163  (class class class)co 5814   e. cmpo 5816   1stc1st 6076   2ndc2nd 6077   supcsup 6914   RR*cxr 7890   < clt 7891   RR+crp 9538   *Metcxmet 12327   MetOpencmopn 12332   CnP ccnp 12533   tX ctx 12599

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-map 6584  df-sup 6916  df-inf 6917  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-xneg 9657  df-xadd 9658  df-seqfrec 10323  df-exp 10397  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-topgen 12319  df-psmet 12334  df-xmet 12335  df-bl 12337  df-mopn 12338  df-top 12343  df-topon 12356  df-bases 12388  df-cnp 12536  df-tx 12600

This theorem is referenced by:  txmetcn  12866  limccnp2cntop  12993