Theorem xpsdsval 20612

Description: Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
xpsds.t	|- T = ( R X.s S )
xpsds.x	|- X = ( Base ` R )
xpsds.y	|- Y = ( Base ` S )
xpsds.1	|- ( ph -> R e. V )
xpsds.2	|- ( ph -> S e. W )
xpsds.p	|- P = ( dist ` T )
xpsds.m	|- M = ( ( dist ` R ) |` ( X X. X ) )
xpsds.n	|- N = ( ( dist ` S ) |` ( Y X. Y ) )
xpsds.3	|- ( ph -> M e. ( *Met ` X ) )
xpsds.4	|- ( ph -> N e. ( *Met ` Y ) )
xpsds.a	|- ( ph -> A e. X )
xpsds.b	|- ( ph -> B e. Y )
xpsds.c	|- ( ph -> C e. X )
xpsds.d	|- ( ph -> D e. Y )

Assertion

Ref	Expression
xpsdsval	|- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Proof of Theorem xpsdsval

Dummy variables x k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsds.t	. . . . 5 |- T = ( R X.s S )
2		xpsds.x	. . . . 5 |- X = ( Base ` R )
3		xpsds.y	. . . . 5 |- Y = ( Base ` S )
4		xpsds.1	. . . . 5 |- ( ph -> R e. V )
5		xpsds.2	. . . . 5 |- ( ph -> S e. W )
6		eqid 2460	. . . . 5 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2460	. . . . 5 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2460	. . . . 5 |- ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s `' ( { R } +c { S } ) )
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 14816	. . . 4 |- ( ph -> T = ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) "s ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8	xpslem 14817	. . . 4 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 14815	. . . . 5 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
12		f1ocnv 5819	. . . . 5 |- ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -> `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -1-1-onto-> ( X X. Y ) )
13	11, 12	mp1i 12	. . . 4 |- ( ph -> `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -1-1-onto-> ( X X. Y ) )
14		ovex 6300	. . . . 5 |- ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V
15	14	a1i 11	. . . 4 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V )
16		eqid 2460	. . . 4 |- ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) = ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
17		xpsds.p	. . . 4 |- P = ( dist ` T )
18		xpsds.m	. . . . . 6 |- M = ( ( dist ` R ) |` ( X X. X ) )
19		xpsds.n	. . . . . 6 |- N = ( ( dist ` S ) |` ( Y X. Y ) )
20		xpsds.3	. . . . . 6 |- ( ph -> M e. ( *Met ` X ) )
21		xpsds.4	. . . . . 6 |- ( ph -> N e. ( *Met ` Y ) )
22	1, 2, 3, 4, 5, 17, 18, 19, 20, 21	xpsxmetlem 20610	. . . . 5 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
23		ssid 3516	. . . . 5 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
24		xmetres2 20592	. . . . 5 |- ( ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) /\ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) -> ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
25	22, 23, 24	sylancl 662	. . . 4 |- ( ph -> ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
26		df-ov 6278	. . . . . 6 |- ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. )
27		xpsds.a	. . . . . . 7 |- ( ph -> A e. X )
28		xpsds.b	. . . . . . 7 |- ( ph -> B e. Y )
29	6	xpsfval 14811	. . . . . . 7 |- ( ( A e. X /\ B e. Y ) -> ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = `' ( { A } +c { B } ) )
30	27, 28, 29	syl2anc 661	. . . . . 6 |- ( ph -> ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = `' ( { A } +c { B } ) )
31	26, 30	syl5eqr 2515	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) )
32		opelxpi 5023	. . . . . . 7 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
33	27, 28, 32	syl2anc 661	. . . . . 6 |- ( ph -> <. A , B >. e. ( X X. Y ) )
34		f1of 5807	. . . . . . . 8 |- ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -> ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) --> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
35	11, 34	ax-mp 5	. . . . . . 7 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) --> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
36	35	ffvelrni 6011	. . . . . 6 |- ( <. A , B >. e. ( X X. Y ) -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
37	33, 36	syl 16	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
38	31, 37	eqeltrrd 2549	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
39		df-ov 6278	. . . . . 6 |- ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. )
40		xpsds.c	. . . . . . 7 |- ( ph -> C e. X )
41		xpsds.d	. . . . . . 7 |- ( ph -> D e. Y )
42	6	xpsfval 14811	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = `' ( { C } +c { D } ) )
43	40, 41, 42	syl2anc 661	. . . . . 6 |- ( ph -> ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = `' ( { C } +c { D } ) )
44	39, 43	syl5eqr 2515	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) )
45		opelxpi 5023	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> <. C , D >. e. ( X X. Y ) )
46	40, 41, 45	syl2anc 661	. . . . . 6 |- ( ph -> <. C , D >. e. ( X X. Y ) )
47	35	ffvelrni 6011	. . . . . 6 |- ( <. C , D >. e. ( X X. Y ) -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
48	46, 47	syl 16	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
49	44, 48	eqeltrrd 2549	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
50	9, 10, 13, 15, 16, 17, 25, 38, 49	imasdsf1o 20605	. . 3 |- ( ph -> ( ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) P ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) ) = ( `' ( { A } +c { B } ) ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) `' ( { C } +c { D } ) ) )
51	38, 49	ovresd 6418	. . 3 |- ( ph -> ( `' ( { A } +c { B } ) ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) `' ( { C } +c { D } ) ) = ( `' ( { A } +c { B } ) ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) `' ( { C } +c { D } ) ) )
52	50, 51	eqtrd 2501	. 2 |- ( ph -> ( ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) P ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) ) = ( `' ( { A } +c { B } ) ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) `' ( { C } +c { D } ) ) )
53		f1ocnvfv 6163	. . . . 5 |- ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) /\ <. A , B >. e. ( X X. Y ) ) -> ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) = <. A , B >. ) )
54	11, 33, 53	sylancr 663	. . . 4 |- ( ph -> ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) = <. A , B >. ) )
55	31, 54	mpd 15	. . 3 |- ( ph -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) = <. A , B >. )
56		f1ocnvfv 6163	. . . . 5 |- ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) /\ <. C , D >. e. ( X X. Y ) ) -> ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) = <. C , D >. ) )
57	11, 46, 56	sylancr 663	. . . 4 |- ( ph -> ( ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) = <. C , D >. ) )
58	44, 57	mpd 15	. . 3 |- ( ph -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) = <. C , D >. )
59	55, 58	oveq12d 6293	. 2 |- ( ph -> ( ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) P ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) ) = ( <. A , B >. P <. C , D >. ) )
60		eqid 2460	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
61		fvex 5867	. . . . 5 |- (Scalar ` R ) e. _V
62	61	a1i 11	. . . 4 |- ( ph -> (Scalar ` R ) e. _V )
63		2on 7128	. . . . 5 |- 2o e. On
64	63	a1i 11	. . . 4 |- ( ph -> 2o e. On )
65		xpscfn 14803	. . . . 5 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
66	4, 5, 65	syl2anc 661	. . . 4 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
67	38, 10	eleqtrd 2550	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
68	49, 10	eleqtrd 2550	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
69		eqid 2460	. . . 4 |- ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
70	8, 60, 62, 64, 66, 67, 68, 69	prdsdsval 14722	. . 3 |- ( ph -> ( `' ( { A } +c { B } ) ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) `' ( { C } +c { D } ) ) = sup ( ( ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) u. { 0 } ) , RR* , < ) )
71		df2o3 7133	. . . . . . . . . . 11 |- 2o = { (/) , 1o }
72	71	rexeqi 3056	. . . . . . . . . 10 |- ( E. k e. 2o x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> E. k e. { (/) , 1o } x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) )
73		0ex 4570	. . . . . . . . . . 11 |- (/) e. _V
74		1on 7127	. . . . . . . . . . . 12 |- 1o e. On
75	74	elexi 3116	. . . . . . . . . . 11 |- 1o e. _V
76		fveq2 5857	. . . . . . . . . . . . . 14 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
77	76	fveq2d 5861	. . . . . . . . . . . . 13 |- ( k = (/) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) )
78		fveq2 5857	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` (/) ) )
79		fveq2 5857	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` (/) ) )
80	77, 78, 79	oveq123d 6296	. . . . . . . . . . . 12 |- ( k = (/) -> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) )
81	80	eqeq2d 2474	. . . . . . . . . . 11 |- ( k = (/) -> ( x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) ) )
82		fveq2 5857	. . . . . . . . . . . . . 14 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
83	82	fveq2d 5861	. . . . . . . . . . . . 13 |- ( k = 1o -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) )
84		fveq2 5857	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` 1o ) )
85		fveq2 5857	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` 1o ) )
86	83, 84, 85	oveq123d 6296	. . . . . . . . . . . 12 |- ( k = 1o -> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) )
87	86	eqeq2d 2474	. . . . . . . . . . 11 |- ( k = 1o -> ( x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) ) )
88	73, 75, 81, 87	rexpr 4074	. . . . . . . . . 10 |- ( E. k e. { (/) , 1o } x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) \/ x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) ) )
89	72, 88	bitri 249	. . . . . . . . 9 |- ( E. k e. 2o x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) \/ x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) ) )
90		xpsc0 14804	. . . . . . . . . . . . . . 15 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
91	4, 90	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
92	91	fveq2d 5861	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) = ( dist ` R ) )
93		xpsc0 14804	. . . . . . . . . . . . . 14 |- ( A e. X -> ( `' ( { A } +c { B } ) ` (/) ) = A )
94	27, 93	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` (/) ) = A )
95		xpsc0 14804	. . . . . . . . . . . . . 14 |- ( C e. X -> ( `' ( { C } +c { D } ) ` (/) ) = C )
96	40, 95	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` (/) ) = C )
97	92, 94, 96	oveq123d 6296	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A ( dist ` R ) C ) )
98	18	oveqi 6288	. . . . . . . . . . . . 13 |- ( A M C ) = ( A ( ( dist ` R ) |` ( X X. X ) ) C )
99	27, 40	ovresd 6418	. . . . . . . . . . . . 13 |- ( ph -> ( A ( ( dist ` R ) |` ( X X. X ) ) C ) = ( A ( dist ` R ) C ) )
100	98, 99	syl5eq 2513	. . . . . . . . . . . 12 |- ( ph -> ( A M C ) = ( A ( dist ` R ) C ) )
101	97, 100	eqtr4d 2504	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A M C ) )
102	101	eqeq2d 2474	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) <-> x = ( A M C ) ) )
103		xpsc1 14805	. . . . . . . . . . . . . . 15 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
104	5, 103	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
105	104	fveq2d 5861	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) = ( dist ` S ) )
106		xpsc1 14805	. . . . . . . . . . . . . 14 |- ( B e. Y -> ( `' ( { A } +c { B } ) ` 1o ) = B )
107	28, 106	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` 1o ) = B )
108		xpsc1 14805	. . . . . . . . . . . . . 14 |- ( D e. Y -> ( `' ( { C } +c { D } ) ` 1o ) = D )
109	41, 108	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` 1o ) = D )
110	105, 107, 109	oveq123d 6296	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B ( dist ` S ) D ) )
111	19	oveqi 6288	. . . . . . . . . . . . 13 |- ( B N D ) = ( B ( ( dist ` S ) |` ( Y X. Y ) ) D )
112	28, 41	ovresd 6418	. . . . . . . . . . . . 13 |- ( ph -> ( B ( ( dist ` S ) |` ( Y X. Y ) ) D ) = ( B ( dist ` S ) D ) )
113	111, 112	syl5eq 2513	. . . . . . . . . . . 12 |- ( ph -> ( B N D ) = ( B ( dist ` S ) D ) )
114	110, 113	eqtr4d 2504	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B N D ) )
115	114	eqeq2d 2474	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) <-> x = ( B N D ) ) )
116	102, 115	orbi12d 709	. . . . . . . . 9 |- ( ph -> ( ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) \/ x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) ) <-> ( x = ( A M C ) \/ x = ( B N D ) ) ) )
117	89, 116	syl5bb 257	. . . . . . . 8 |- ( ph -> ( E. k e. 2o x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) <-> ( x = ( A M C ) \/ x = ( B N D ) ) ) )
118		vex 3109	. . . . . . . . 9 |- x e. _V
119		eqid 2460	. . . . . . . . . 10 |- ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) = ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) )
120	119	elrnmpt 5240	. . . . . . . . 9 |- ( x e. _V -> ( x e. ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) <-> E. k e. 2o x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) )
121	118, 120	ax-mp 5	. . . . . . . 8 |- ( x e. ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) <-> E. k e. 2o x = ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) )
122	118	elpr 4038	. . . . . . . 8 |- ( x e. { ( A M C ) , ( B N D ) } <-> ( x = ( A M C ) \/ x = ( B N D ) ) )
123	117, 121, 122	3bitr4g 288	. . . . . . 7 |- ( ph -> ( x e. ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) <-> x e. { ( A M C ) , ( B N D ) } ) )
124	123	eqrdv 2457	. . . . . 6 |- ( ph -> ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) = { ( A M C ) , ( B N D ) } )
125	124	uneq1d 3650	. . . . 5 |- ( ph -> ( ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) u. { 0 } ) = ( { ( A M C ) , ( B N D ) } u. { 0 } ) )
126		uncom 3641	. . . . 5 |- ( { ( A M C ) , ( B N D ) } u. { 0 } ) = ( { 0 } u. { ( A M C ) , ( B N D ) } )
127	125, 126	syl6eq 2517	. . . 4 |- ( ph -> ( ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) u. { 0 } ) = ( { 0 } u. { ( A M C ) , ( B N D ) } ) )
128	127	supeq1d 7895	. . 3 |- ( ph -> sup ( ( ran ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( dist ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) u. { 0 } ) , RR* , < ) = sup ( ( { 0 } u. { ( A M C ) , ( B N D ) } ) , RR* , < ) )
129		0xr 9629	. . . . . 6 |- 0 e. RR*
130	129	a1i 11	. . . . 5 |- ( ph -> 0 e. RR* )
131	130	snssd 4165	. . . 4 |- ( ph -> { 0 } C_ RR* )
132		xmetcl 20562	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> ( A M C ) e. RR* )
133	20, 27, 40, 132	syl3anc 1223	. . . . 5 |- ( ph -> ( A M C ) e. RR* )
134		xmetcl 20562	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ B e. Y /\ D e. Y ) -> ( B N D ) e. RR* )
135	21, 28, 41, 134	syl3anc 1223	. . . . 5 |- ( ph -> ( B N D ) e. RR* )
136		prssi 4176	. . . . 5 |- ( ( ( A M C ) e. RR* /\ ( B N D ) e. RR* ) -> { ( A M C ) , ( B N D ) } C_ RR* )
137	133, 135, 136	syl2anc 661	. . . 4 |- ( ph -> { ( A M C ) , ( B N D ) } C_ RR* )
138		xrltso 11336	. . . . . 6 |- < Or RR*
139		supsn 7919	. . . . . 6 |- ( ( < Or RR* /\ 0 e. RR* ) -> sup ( { 0 } , RR* , < ) = 0 )
140	138, 129, 139	mp2an 672	. . . . 5 |- sup ( { 0 } , RR* , < ) = 0
141		supxrcl 11495	. . . . . . 7 |- ( { ( A M C ) , ( B N D ) } C_ RR* -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
142	137, 141	syl 16	. . . . . 6 |- ( ph -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
143		xmetge0 20575	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> 0 <_ ( A M C ) )
144	20, 27, 40, 143	syl3anc 1223	. . . . . 6 |- ( ph -> 0 <_ ( A M C ) )
145		ovex 6300	. . . . . . . 8 |- ( A M C ) e. _V
146	145	prid1 4128	. . . . . . 7 |- ( A M C ) e. { ( A M C ) , ( B N D ) }
147		supxrub 11505	. . . . . . 7 |- ( ( { ( A M C ) , ( B N D ) } C_ RR* /\ ( A M C ) e. { ( A M C ) , ( B N D ) } ) -> ( A M C ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
148	137, 146, 147	sylancl 662	. . . . . 6 |- ( ph -> ( A M C ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
149	130, 133, 142, 144, 148	xrletrd 11354	. . . . 5 |- ( ph -> 0 <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
150	140, 149	syl5eqbr 4473	. . . 4 |- ( ph -> sup ( { 0 } , RR* , < ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
151		supxrun 11496	. . . 4 |- ( ( { 0 } C_ RR* /\ { ( A M C ) , ( B N D ) } C_ RR* /\ sup ( { 0 } , RR* , < ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) ) -> sup ( ( { 0 } u. { ( A M C ) , ( B N D ) } ) , RR* , < ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
152	131, 137, 150, 151	syl3anc 1223	. . 3 |- ( ph -> sup ( ( { 0 } u. { ( A M C ) , ( B N D ) } ) , RR* , < ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
153	70, 128, 152	3eqtrd 2505	. 2 |- ( ph -> ( `' ( { A } +c { B } ) ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) `' ( { C } +c { D } ) ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
154	52, 59, 153	3eqtr3d 2509	1 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   = wceq 1374   e. wcel 1762   E.wrex 2808   _Vcvv 3106   u. cun 3467   C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   <.cop 4026   class class class wbr 4440   |-> cmpt 4498   Or wor 4792   Oncon0 4871   X. cxp 4990   `'ccnv 4991   ran crn 4993   |` cres 4994   Fn wfn 5574   -->wf 5575   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   |-> cmpt2 6277   1oc1o 7113   2oc2o 7114   supcsup 7889   +c ccda 8536   0cc0 9481   RR*cxr 9616   < clt 9617   <_ cle 9618   Basecbs 14479  Scalarcsca 14547   distcds 14553   X__scprds 14690   X._s cxps 14750   *Metcxmt 18167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-xrs 14746  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-xmet 18176

This theorem is referenced by:  tmsxpsval  20769