Theorem xpsdsval 21176

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 2402	. . . . 5 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2402	. . . . 5 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2402	. . . . 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 15186	. . . 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 15187	. . . 4 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 15185	. . . . 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 5811	. . . . 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		ovex 6306	. . . . 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 2402	. . . 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 21174	. . . . 5 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
23		ssid 3461	. . . . 5 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
24		xmetres2 21156	. . . . 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 660	. . . 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 6281	. . . . . 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 15181	. . . . . . 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 659	. . . . . 6 |- ( ph -> ( A ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) B ) = `' ( { A } +c { B } ) )
31	26, 30	syl5eqr 2457	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. A , B >. ) = `' ( { A } +c { B } ) )
32		opelxpi 4855	. . . . . . 7 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
33	27, 28, 32	syl2anc 659	. . . . . 6 |- ( ph -> <. A , B >. e. ( X X. Y ) )
34		f1of 5799	. . . . . . . 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 6008	. . . . . 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 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 } ) ) )
38	31, 37	eqeltrrd 2491	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) )
39		df-ov 6281	. . . . . 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 15181	. . . . . . 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 659	. . . . . 6 |- ( ph -> ( C ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) D ) = `' ( { C } +c { D } ) )
44	39, 43	syl5eqr 2457	. . . . 5 |- ( ph -> ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ` <. C , D >. ) = `' ( { C } +c { D } ) )
45		opelxpi 4855	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> <. C , D >. e. ( X X. Y ) )
46	40, 41, 45	syl2anc 659	. . . . . 6 |- ( ph -> <. C , D >. e. ( X X. Y ) )
47	35	ffvelrni 6008	. . . . . 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 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 } ) ) )
49	44, 48	eqeltrrd 2491	. . . 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 21169	. . 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 6424	. . 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 2443	. 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 6165	. . . . 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 661	. . . 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 6165	. . . . 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 661	. . . 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 6296	. 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 2402	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) )
61		fvex 5859	. . . . 5 |- (Scalar ` R ) e. _V
62	61	a1i 11	. . . 4 |- ( ph -> (Scalar ` R ) e. _V )
63		2on 7175	. . . . 5 |- 2o e. On
64	63	a1i 11	. . . 4 |- ( ph -> 2o e. On )
65		xpscfn 15173	. . . . 5 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
66	4, 5, 65	syl2anc 659	. . . 4 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
67	38, 10	eleqtrd 2492	. . . 4 |- ( ph -> `' ( { A } +c { B } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
68	49, 10	eleqtrd 2492	. . . 4 |- ( ph -> `' ( { C } +c { D } ) e. ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
69		eqid 2402	. . . 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 15092	. . 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 7180	. . . . . . . . . . 11 |- 2o = { (/) , 1o }
72	71	rexeqi 3009	. . . . . . . . . 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 4526	. . . . . . . . . . 11 |- (/) e. _V
74		1on 7174	. . . . . . . . . . . 12 |- 1o e. On
75	74	elexi 3069	. . . . . . . . . . 11 |- 1o e. _V
76		fveq2 5849	. . . . . . . . . . . . . 14 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
77	76	fveq2d 5853	. . . . . . . . . . . . 13 |- ( k = (/) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) )
78		fveq2 5849	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` (/) ) )
79		fveq2 5849	. . . . . . . . . . . . 13 |- ( k = (/) -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` (/) ) )
80	77, 78, 79	oveq123d 6299	. . . . . . . . . . . 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 2416	. . . . . . . . . . 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 5849	. . . . . . . . . . . . . 14 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
83	82	fveq2d 5853	. . . . . . . . . . . . 13 |- ( k = 1o -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) )
84		fveq2 5849	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { A } +c { B } ) ` k ) = ( `' ( { A } +c { B } ) ` 1o ) )
85		fveq2 5849	. . . . . . . . . . . . 13 |- ( k = 1o -> ( `' ( { C } +c { D } ) ` k ) = ( `' ( { C } +c { D } ) ` 1o ) )
86	83, 84, 85	oveq123d 6299	. . . . . . . . . . . 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 2416	. . . . . . . . . . 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 4026	. . . . . . . . . 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 15174	. . . . . . . . . . . . . . 15 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
91	4, 90	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
92	91	fveq2d 5853	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) = ( dist ` R ) )
93		xpsc0 15174	. . . . . . . . . . . . . 14 |- ( A e. X -> ( `' ( { A } +c { B } ) ` (/) ) = A )
94	27, 93	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` (/) ) = A )
95		xpsc0 15174	. . . . . . . . . . . . . 14 |- ( C e. X -> ( `' ( { C } +c { D } ) ` (/) ) = C )
96	40, 95	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` (/) ) = C )
97	92, 94, 96	oveq123d 6299	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A ( dist ` R ) C ) )
98	18	oveqi 6291	. . . . . . . . . . . . 13 |- ( A M C ) = ( A ( ( dist ` R ) |` ( X X. X ) ) C )
99	27, 40	ovresd 6424	. . . . . . . . . . . . 13 |- ( ph -> ( A ( ( dist ` R ) |` ( X X. X ) ) C ) = ( A ( dist ` R ) C ) )
100	98, 99	syl5eq 2455	. . . . . . . . . . . 12 |- ( ph -> ( A M C ) = ( A ( dist ` R ) C ) )
101	97, 100	eqtr4d 2446	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) = ( A M C ) )
102	101	eqeq2d 2416	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` (/) ) ( dist ` ( `' ( { R } +c { S } ) ` (/) ) ) ( `' ( { C } +c { D } ) ` (/) ) ) <-> x = ( A M C ) ) )
103		xpsc1 15175	. . . . . . . . . . . . . . 15 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
104	5, 103	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
105	104	fveq2d 5853	. . . . . . . . . . . . 13 |- ( ph -> ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) = ( dist ` S ) )
106		xpsc1 15175	. . . . . . . . . . . . . 14 |- ( B e. Y -> ( `' ( { A } +c { B } ) ` 1o ) = B )
107	28, 106	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { A } +c { B } ) ` 1o ) = B )
108		xpsc1 15175	. . . . . . . . . . . . . 14 |- ( D e. Y -> ( `' ( { C } +c { D } ) ` 1o ) = D )
109	41, 108	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( { C } +c { D } ) ` 1o ) = D )
110	105, 107, 109	oveq123d 6299	. . . . . . . . . . . 12 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B ( dist ` S ) D ) )
111	19	oveqi 6291	. . . . . . . . . . . . 13 |- ( B N D ) = ( B ( ( dist ` S ) |` ( Y X. Y ) ) D )
112	28, 41	ovresd 6424	. . . . . . . . . . . . 13 |- ( ph -> ( B ( ( dist ` S ) |` ( Y X. Y ) ) D ) = ( B ( dist ` S ) D ) )
113	111, 112	syl5eq 2455	. . . . . . . . . . . 12 |- ( ph -> ( B N D ) = ( B ( dist ` S ) D ) )
114	110, 113	eqtr4d 2446	. . . . . . . . . . 11 |- ( ph -> ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) = ( B N D ) )
115	114	eqeq2d 2416	. . . . . . . . . 10 |- ( ph -> ( x = ( ( `' ( { A } +c { B } ) ` 1o ) ( dist ` ( `' ( { R } +c { S } ) ` 1o ) ) ( `' ( { C } +c { D } ) ` 1o ) ) <-> x = ( B N D ) ) )
116	102, 115	orbi12d 708	. . . . . . . . 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 3062	. . . . . . . . 9 |- x e. _V
119		eqid 2402	. . . . . . . . . 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 5070	. . . . . . . . 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 3990	. . . . . . . 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 2399	. . . . . 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 3596	. . . . 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 3587	. . . . 5 |- ( { ( A M C ) , ( B N D ) } u. { 0 } ) = ( { 0 } u. { ( A M C ) , ( B N D ) } )
127	125, 126	syl6eq 2459	. . . 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 7939	. . 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 9670	. . . . . 6 |- 0 e. RR*
130	129	a1i 11	. . . . 5 |- ( ph -> 0 e. RR* )
131	130	snssd 4117	. . . 4 |- ( ph -> { 0 } C_ RR* )
132		xmetcl 21126	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> ( A M C ) e. RR* )
133	20, 27, 40, 132	syl3anc 1230	. . . . 5 |- ( ph -> ( A M C ) e. RR* )
134		xmetcl 21126	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ B e. Y /\ D e. Y ) -> ( B N D ) e. RR* )
135	21, 28, 41, 134	syl3anc 1230	. . . . 5 |- ( ph -> ( B N D ) e. RR* )
136		prssi 4128	. . . . 5 |- ( ( ( A M C ) e. RR* /\ ( B N D ) e. RR* ) -> { ( A M C ) , ( B N D ) } C_ RR* )
137	133, 135, 136	syl2anc 659	. . . 4 |- ( ph -> { ( A M C ) , ( B N D ) } C_ RR* )
138		xrltso 11400	. . . . . 6 |- < Or RR*
139		supsn 7964	. . . . . 6 |- ( ( < Or RR* /\ 0 e. RR* ) -> sup ( { 0 } , RR* , < ) = 0 )
140	138, 129, 139	mp2an 670	. . . . 5 |- sup ( { 0 } , RR* , < ) = 0
141		supxrcl 11559	. . . . . . 7 |- ( { ( A M C ) , ( B N D ) } C_ RR* -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
142	137, 141	syl 17	. . . . . 6 |- ( ph -> sup ( { ( A M C ) , ( B N D ) } , RR* , < ) e. RR* )
143		xmetge0 21139	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ A e. X /\ C e. X ) -> 0 <_ ( A M C ) )
144	20, 27, 40, 143	syl3anc 1230	. . . . . 6 |- ( ph -> 0 <_ ( A M C ) )
145		ovex 6306	. . . . . . . 8 |- ( A M C ) e. _V
146	145	prid1 4080	. . . . . . 7 |- ( A M C ) e. { ( A M C ) , ( B N D ) }
147		supxrub 11569	. . . . . . 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 660	. . . . . 6 |- ( ph -> ( A M C ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
149	130, 133, 142, 144, 148	xrletrd 11418	. . . . 5 |- ( ph -> 0 <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
150	140, 149	syl5eqbr 4428	. . . 4 |- ( ph -> sup ( { 0 } , RR* , < ) <_ sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
151		supxrun 11560	. . . 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 1230	. . 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 2447	. 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 2451	1 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   = wceq 1405   e. wcel 1842   E.wrex 2755   _Vcvv 3059   u. cun 3412   C_ wss 3414   (/)c0 3738   {csn 3972   {cpr 3974   <.cop 3978   class class class wbr 4395   |-> cmpt 4453   Or wor 4743   X. cxp 4821   `'ccnv 4822   ran crn 4824   |` cres 4825   Oncon0 5410   Fn wfn 5564   -->wf 5565   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   |-> cmpt2 6280   1oc1o 7160   2oc2o 7161   supcsup 7934   +c ccda 8579   0cc0 9522   RR*cxr 9657   < clt 9658   <_ cle 9659   Basecbs 14841  Scalarcsca 14912   distcds 14918   X__scprds 15060   X._s cxps 15120   *Metcxmt 18723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-xrs 15116  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-xmet 18732

This theorem is referenced by:  tmsxpsval  21333