Theorem xpsdsval 22186

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 2622	. . . . 5 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2622	. . . . 5 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2622	. . . . 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 16232	. . . 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 16233	. . . 4 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 16231	. . . . 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 6149	. . . . 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 13	. . . 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		ovexd 6680	. . . 4 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V )
15		eqid 2622	. . . 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 } ) ) ) )
16		xpsds.p	. . . 4 |- P = ( dist ` T )
17		xpsds.m	. . . . . 6 |- M = ( ( dist ` R ) |` ( X X. X ) )
18		xpsds.n	. . . . . 6 |- N = ( ( dist ` S ) |` ( Y X. Y ) )
19		xpsds.3	. . . . . 6 |- ( ph -> M e. ( *Met ` X ) )
20		xpsds.4	. . . . . 6 |- ( ph -> N e. ( *Met ` Y ) )
21	1, 2, 3, 4, 5, 16, 17, 18, 19, 20	xpsxmetlem 22184	. . . . 5 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
22		ssid 3624	. . . . 5 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
23		xmetres2 22166	. . . . 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 } ) ) ) )
24	21, 22, 23	sylancl 694	. . . 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 } ) ) ) )
25		df-ov 6653	. . . . . 6 |- ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. )
26		xpsds.a	. . . . . . 7 |- ( ph -> A e. X )
27		xpsds.b	. . . . . . 7 |- ( ph -> B e. Y )
28	6	xpsfval 16227	. . . . . . 7 |- ( ( A e. X /\ B e. Y ) -> ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = `' ( { A } +c { B } ) )
29	26, 27, 28	syl2anc 693	. . . . . 6 |- ( ph -> ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = `' ( { A } +c { B } ) )
30	25, 29	syl5eqr 2670	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) )
31		opelxpi 5148	. . . . . . 7 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
32	26, 27, 31	syl2anc 693	. . . . . 6 |- ( ph -> <. A , B >. e. ( X X. Y ) )
33		f1of 6137	. . . . . . . 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 } ) ) )
34	11, 33	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 } ) )
35	34	ffvelrni 6358	. . . . . 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 } ) ) )
36	32, 35	syl 17	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
37	30, 36	eqeltrrd 2702	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
38		df-ov 6653	. . . . . 6 |- ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. )
39		xpsds.c	. . . . . . 7 |- ( ph -> C e. X )
40		xpsds.d	. . . . . . 7 |- ( ph -> D e. Y )
41	6	xpsfval 16227	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = `' ( { C } +c { D } ) )
42	39, 40, 41	syl2anc 693	. . . . . 6 |- ( ph -> ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = `' ( { C } +c { D } ) )
43	38, 42	syl5eqr 2670	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) )
44		opelxpi 5148	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> <. C , D >. e. ( X X. Y ) )
45	39, 40, 44	syl2anc 693	. . . . . 6 |- ( ph -> <. C , D >. e. ( X X. Y ) )
46	34	ffvelrni 6358	. . . . . 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 } ) ) )
47	45, 46	syl 17	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
48	43, 47	eqeltrrd 2702	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
49	9, 10, 13, 14, 15, 16, 24, 37, 48	imasdsf1o 22179	. . 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 } ) ) )
50	37, 48	ovresd 6801	. . 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 } ) ) )
51	49, 50	eqtrd 2656	. 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 } ) ) )
52		f1ocnvfv 6534	. . . . 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 >. ) )
53	11, 32, 52	sylancr 695	. . . 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 >. ) )
54	30, 53	mpd 15	. . 3 |- ( ph -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { A } +c { B } ) ) = <. A , B >. )
55		f1ocnvfv 6534	. . . . 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 >. ) )
56	11, 45, 55	sylancr 695	. . . 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 >. ) )
57	43, 56	mpd 15	. . 3 |- ( ph -> ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` `' ( { C } +c { D } ) ) = <. C , D >. )
58	54, 57	oveq12d 6668	. 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 >. ) )
59		eqid 2622	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
60		fvexd 6203	. . . 4 |- ( ph -> (Scalar ` R ) e. _V )
61		2on 7568	. . . . 5 |- 2o e. On
62	61	a1i 11	. . . 4 |- ( ph -> 2o e. On )
63		xpscfn 16219	. . . . 5 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
64	4, 5, 63	syl2anc 693	. . . 4 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
65	37, 10	eleqtrd 2703	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
66	48, 10	eleqtrd 2703	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
67		eqid 2622	. . . 4 |- ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
68	8, 59, 60, 62, 64, 65, 66, 67	prdsdsval 16138	. . 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* , < ) )
69		df2o3 7573	. . . . . . . . . . 11 |- 2o = { (/) , 1o }
70	69	rexeqi 3143	. . . . . . . . . 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 ) ) )
71		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
72		1on 7567	. . . . . . . . . . . 12 |- 1o e. On
73	72	elexi 3213	. . . . . . . . . . 11 |- 1o e. _V
74		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
75	74	fveq2d 6195	. . . . . . . . . . . . 13 |- ( k = (/) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) )
76		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` (/) ) )
77		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` (/) ) )
78	75, 76, 77	oveq123d 6671	. . . . . . . . . . . 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 } ) ` (/) ) ) )
79	78	eqeq2d 2632	. . . . . . . . . . 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 } ) ` (/) ) ) ) )
80		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
81	80	fveq2d 6195	. . . . . . . . . . . . 13 |- ( k = 1o -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) )
82		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` 1o ) )
83		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` 1o ) )
84	81, 82, 83	oveq123d 6671	. . . . . . . . . . . 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 ) ) )
85	84	eqeq2d 2632	. . . . . . . . . . 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 ) ) ) )
86	71, 73, 79, 85	rexpr 4239	. . . . . . . . . 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 ) ) ) )
87	70, 86	bitri 264	. . . . . . . . 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 ) ) ) )
88		xpsc0 16220	. . . . . . . . . . . . . . 15 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
89	4, 88	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
90	89	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) = ( dist ` R ) )
91		xpsc0 16220	. . . . . . . . . . . . . 14 |- ( A e. X -> ( `' ( { A } +c { B } ) ` (/) ) = A )
92	26, 91	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` (/) ) = A )
93		xpsc0 16220	. . . . . . . . . . . . . 14 |- ( C e. X -> ( `' ( { C } +c { D } ) ` (/) ) = C )
94	39, 93	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` (/) ) = C )
95	90, 92, 94	oveq123d 6671	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A ( dist ` R ) C ) )
96	17	oveqi 6663	. . . . . . . . . . . . 13 |- ( A M C ) = ( A ( ( dist ` R ) |` ( X X. X ) ) C )
97	26, 39	ovresd 6801	. . . . . . . . . . . . 13 |- ( ph -> ( A ( ( dist ` R ) |` ( X X. X ) ) C ) = ( A ( dist ` R ) C ) )
98	96, 97	syl5eq 2668	. . . . . . . . . . . 12 |- ( ph -> ( A M C ) = ( A ( dist ` R ) C ) )
99	95, 98	eqtr4d 2659	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A M C ) )
100	99	eqeq2d 2632	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) <-> x = ( A M C ) ) )
101		xpsc1 16221	. . . . . . . . . . . . . . 15 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
102	5, 101	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
103	102	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) = ( dist ` S ) )
104		xpsc1 16221	. . . . . . . . . . . . . 14 |- ( B e. Y -> ( `' ( { A } +c { B } ) ` 1o ) = B )
105	27, 104	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` 1o ) = B )
106		xpsc1 16221	. . . . . . . . . . . . . 14 |- ( D e. Y -> ( `' ( { C } +c { D } ) ` 1o ) = D )
107	40, 106	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` 1o ) = D )
108	103, 105, 107	oveq123d 6671	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B ( dist ` S ) D ) )
109	18	oveqi 6663	. . . . . . . . . . . . 13 |- ( B N D ) = ( B ( ( dist ` S ) |` ( Y X. Y ) ) D )
110	27, 40	ovresd 6801	. . . . . . . . . . . . 13 |- ( ph -> ( B ( ( dist ` S ) |` ( Y X. Y ) ) D ) = ( B ( dist ` S ) D ) )
111	109, 110	syl5eq 2668	. . . . . . . . . . . 12 |- ( ph -> ( B N D ) = ( B ( dist ` S ) D ) )
112	108, 111	eqtr4d 2659	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B N D ) )
113	112	eqeq2d 2632	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) <-> x = ( B N D ) ) )
114	100, 113	orbi12d 746	. . . . . . . . 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 ) ) ) )
115	87, 114	syl5bb 272	. . . . . . . 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 ) ) ) )
116		vex 3203	. . . . . . . . 9 |- x e. _V
117		eqid 2622	. . . . . . . . . 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 ) ) )
118	117	elrnmpt 5372	. . . . . . . . 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 ) ) ) )
119	116, 118	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 ) ) )
120	116	elpr 4198	. . . . . . . 8 |- ( x e. { ( A M C ) , ( B N D ) } <-> ( x = ( A M C ) \/ x = ( B N D ) ) )
121	115, 119, 120	3bitr4g 303	. . . . . . 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 ) } ) )
122	121	eqrdv 2620	. . . . . 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 ) } )
123	122	uneq1d 3766	. . . . 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 } ) )
124		uncom 3757	. . . . 5 |- ( { ( A M C ) , ( B N D ) } u. { 0 } ) = ( { 0 } u. { ( A M C ) , ( B N D ) } )
125	123, 124	syl6eq 2672	. . . 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 ) } ) )
126	125	supeq1d 8352	. . 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* , < ) )
127		0xr 10086	. . . . . 6 |- 0 e. RR*
128	127	a1i 11	. . . . 5 |- ( ph -> 0 e. RR* )
129	128	snssd 4340	. . . 4 |- ( ph -> { 0 } C_ RR* )
130		xmetcl 22136	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> ( A M C ) e. RR* )
131	19, 26, 39, 130	syl3anc 1326	. . . . 5 |- ( ph -> ( A M C ) e. RR* )
132		xmetcl 22136	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ B e. Y /\ D e. Y ) -> ( B N D ) e. RR* )
133	20, 27, 40, 132	syl3anc 1326	. . . . 5 |- ( ph -> ( B N D ) e. RR* )
134		prssi 4353	. . . . 5 |- ( ( ( A M C ) e. RR* /\ ( B N D ) e. RR* ) -> { ( A M C ) , ( B N D ) } C_ RR* )
135	131, 133, 134	syl2anc 693	. . . 4 |- ( ph -> { ( A M C ) , ( B N D ) } C_ RR* )
136		xrltso 11974	. . . . . 6 |- < Or RR*
137		supsn 8378	. . . . . 6 |- ( ( < Or RR* /\ 0 e. RR* ) -> sup ( { 0 } , RR* , < ) = 0 )
138	136, 127, 137	mp2an 708	. . . . 5 |- sup ( { 0 } , RR* , < ) = 0
139		supxrcl 12145	. . . . . . 7 |- ( { ( A M C ) , ( B N D ) } C_ RR* -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
140	135, 139	syl 17	. . . . . 6 |- ( ph -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
141		xmetge0 22149	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> 0 <_ ( A M C ) )
142	19, 26, 39, 141	syl3anc 1326	. . . . . 6 |- ( ph -> 0 <_ ( A M C ) )
143		ovex 6678	. . . . . . . 8 |- ( A M C ) e. _V
144	143	prid1 4297	. . . . . . 7 |- ( A M C ) e. { ( A M C ) , ( B N D ) }
145		supxrub 12154	. . . . . . 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* , < ) )
146	135, 144, 145	sylancl 694	. . . . . 6 |- ( ph -> ( A M C ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
147	128, 131, 140, 142, 146	xrletrd 11993	. . . . 5 |- ( ph -> 0 <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
148	138, 147	syl5eqbr 4688	. . . 4 |- ( ph -> sup ( { 0 } , RR* , < ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
149		supxrun 12146	. . . 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* , < ) )
150	129, 135, 148, 149	syl3anc 1326	. . 3 |- ( ph -> sup ( ( { 0 } u. { ( A M C ) , ( B N D ) } ) , RR* , < ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
151	68, 126, 150	3eqtrd 2660	. 2 |- ( ph -> ( `' ( { A } +c { B } ) ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) `' ( { C } +c { D } ) ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
152	51, 58, 151	3eqtr3d 2664	1 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   Oncon0 5723   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   supcsup 8346   +c ccda 8989   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   Basecbs 15857  Scalarcsca 15944   distcds 15950   X__scprds 16106   X._s cxps 16166   *Metcxmt 19731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-xrs 16162  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-xmet 19739

This theorem is referenced by:  tmsxpsval  22343