Theorem xpsdsval 20072

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 2451	. . . . 5 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2451	. . . . 5 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2451	. . . . 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 14612	. . . 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 14613	. . . 4 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 14611	. . . . 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 5751	. . . . 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 6215	. . . . 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 2451	. . . 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 20070	. . . . 5 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
23		ssid 3473	. . . . 5 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
24		xmetres2 20052	. . . . 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 6193	. . . . . 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 14607	. . . . . . 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 2506	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) )
32		opelxpi 4969	. . . . . . 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 5739	. . . . . . . 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 5941	. . . . . 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 2540	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
39		df-ov 6193	. . . . . 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 14607	. . . . . . 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 2506	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) )
45		opelxpi 4969	. . . . . . 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 5941	. . . . . 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 2540	. . . 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 20065	. . 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 6331	. . 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 2492	. 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 6084	. . . . 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 6084	. . . . 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 6208	. 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 2451	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
61		fvex 5799	. . . . 5 |- (Scalar ` R ) e. _V
62	61	a1i 11	. . . 4 |- ( ph -> (Scalar ` R ) e. _V )
63		2on 7028	. . . . 5 |- 2o e. On
64	63	a1i 11	. . . 4 |- ( ph -> 2o e. On )
65		xpscfn 14599	. . . . 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 2541	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
68	49, 10	eleqtrd 2541	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
69		eqid 2451	. . . 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 14518	. . 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 7033	. . . . . . . . . . 11 |- 2o = { (/) , 1o }
72	71	rexeqi 3018	. . . . . . . . . 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 4520	. . . . . . . . . . 11 |- (/) e. _V
74		1on 7027	. . . . . . . . . . . 12 |- 1o e. On
75	74	elexi 3078	. . . . . . . . . . 11 |- 1o e. _V
76		fveq2 5789	. . . . . . . . . . . . . 14 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
77	76	fveq2d 5793	. . . . . . . . . . . . 13 |- ( k = (/) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) )
78		fveq2 5789	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` (/) ) )
79		fveq2 5789	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` (/) ) )
80	77, 78, 79	oveq123d 6211	. . . . . . . . . . . 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 2465	. . . . . . . . . . 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 5789	. . . . . . . . . . . . . 14 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
83	82	fveq2d 5793	. . . . . . . . . . . . 13 |- ( k = 1o -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) )
84		fveq2 5789	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` 1o ) )
85		fveq2 5789	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` 1o ) )
86	83, 84, 85	oveq123d 6211	. . . . . . . . . . . 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 2465	. . . . . . . . . . 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 4028	. . . . . . . . . 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 14600	. . . . . . . . . . . . . . 15 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
91	4, 90	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
92	91	fveq2d 5793	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) = ( dist ` R ) )
93		xpsc0 14600	. . . . . . . . . . . . . 14 |- ( A e. X -> ( `' ( { A } +c { B } ) ` (/) ) = A )
94	27, 93	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` (/) ) = A )
95		xpsc0 14600	. . . . . . . . . . . . . 14 |- ( C e. X -> ( `' ( { C } +c { D } ) ` (/) ) = C )
96	40, 95	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` (/) ) = C )
97	92, 94, 96	oveq123d 6211	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A ( dist ` R ) C ) )
98	18	oveqi 6203	. . . . . . . . . . . . 13 |- ( A M C ) = ( A ( ( dist ` R ) |` ( X X. X ) ) C )
99	27, 40	ovresd 6331	. . . . . . . . . . . . 13 |- ( ph -> ( A ( ( dist ` R ) |` ( X X. X ) ) C ) = ( A ( dist ` R ) C ) )
100	98, 99	syl5eq 2504	. . . . . . . . . . . 12 |- ( ph -> ( A M C ) = ( A ( dist ` R ) C ) )
101	97, 100	eqtr4d 2495	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A M C ) )
102	101	eqeq2d 2465	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) <-> x = ( A M C ) ) )
103		xpsc1 14601	. . . . . . . . . . . . . . 15 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
104	5, 103	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
105	104	fveq2d 5793	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) = ( dist ` S ) )
106		xpsc1 14601	. . . . . . . . . . . . . 14 |- ( B e. Y -> ( `' ( { A } +c { B } ) ` 1o ) = B )
107	28, 106	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` 1o ) = B )
108		xpsc1 14601	. . . . . . . . . . . . . 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 6211	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B ( dist ` S ) D ) )
111	19	oveqi 6203	. . . . . . . . . . . . 13 |- ( B N D ) = ( B ( ( dist ` S ) |` ( Y X. Y ) ) D )
112	28, 41	ovresd 6331	. . . . . . . . . . . . 13 |- ( ph -> ( B ( ( dist ` S ) |` ( Y X. Y ) ) D ) = ( B ( dist ` S ) D ) )
113	111, 112	syl5eq 2504	. . . . . . . . . . . 12 |- ( ph -> ( B N D ) = ( B ( dist ` S ) D ) )
114	110, 113	eqtr4d 2495	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B N D ) )
115	114	eqeq2d 2465	. . . . . . . . . 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 3071	. . . . . . . . 9 |- x e. _V
119		eqid 2451	. . . . . . . . . 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 5184	. . . . . . . . 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 3993	. . . . . . . 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 2448	. . . . . 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 3607	. . . . 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 3598	. . . . 5 |- ( { ( A M C ) , ( B N D ) } u. { 0 } ) = ( { 0 } u. { ( A M C ) , ( B N D ) } )
127	125, 126	syl6eq 2508	. . . 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 7797	. . 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 9531	. . . . . 6 |- 0 e. RR*
130	129	a1i 11	. . . . 5 |- ( ph -> 0 e. RR* )
131	130	snssd 4116	. . . 4 |- ( ph -> { 0 } C_ RR* )
132		xmetcl 20022	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> ( A M C ) e. RR* )
133	20, 27, 40, 132	syl3anc 1219	. . . . 5 |- ( ph -> ( A M C ) e. RR* )
134		xmetcl 20022	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ B e. Y /\ D e. Y ) -> ( B N D ) e. RR* )
135	21, 28, 41, 134	syl3anc 1219	. . . . 5 |- ( ph -> ( B N D ) e. RR* )
136		prssi 4127	. . . . 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 11219	. . . . . 6 |- < Or RR*
139		supsn 7820	. . . . . 6 |- ( ( < Or RR* /\ 0 e. RR* ) -> sup ( { 0 } , RR* , < ) = 0 )
140	138, 129, 139	mp2an 672	. . . . 5 |- sup ( { 0 } , RR* , < ) = 0
141		supxrcl 11378	. . . . . . 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 20035	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> 0 <_ ( A M C ) )
144	20, 27, 40, 143	syl3anc 1219	. . . . . 6 |- ( ph -> 0 <_ ( A M C ) )
145		ovex 6215	. . . . . . . 8 |- ( A M C ) e. _V
146	145	prid1 4081	. . . . . . 7 |- ( A M C ) e. { ( A M C ) , ( B N D ) }
147		supxrub 11388	. . . . . . 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 11237	. . . . 5 |- ( ph -> 0 <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
150	140, 149	syl5eqbr 4423	. . . 4 |- ( ph -> sup ( { 0 } , RR* , < ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
151		supxrun 11379	. . . 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 1219	. . 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 2496	. 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 2500	1 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   = wceq 1370   e. wcel 1758   E.wrex 2796   _Vcvv 3068   u. cun 3424   C_ wss 3426   (/)c0 3735   {csn 3975   {cpr 3977   <.cop 3981   class class class wbr 4390   |-> cmpt 4448   Or wor 4738   Oncon0 4817   X. cxp 4936   `'ccnv 4937   ran crn 4939   |` cres 4940   Fn wfn 5511   -->wf 5512   -1-1-onto->wf1o 5515   ` cfv 5516  (class class class)co 6190   |-> cmpt2 6192   1oc1o 7013   2oc2o 7014   supcsup 7791   +c ccda 8437   0cc0 9383   RR*cxr 9518   < clt 9519   <_ cle 9520   Basecbs 14276  Scalarcsca 14343   distcds 14349   X__scprds 14486   X._s cxps 14546   *Metcxmt 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-xrs 14542  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-xmet 17919

This theorem is referenced by:  tmsxpsval  20229