Theorem xmetxp 12676

Description: The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)

Hypotheses

Ref	Expression
xmetxp.p	|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
xmetxp.1	|- ( ph -> M e. ( *Met ` X ) )
xmetxp.2	|- ( ph -> N e. ( *Met ` Y ) )

Assertion

Ref	Expression
xmetxp	|- ( ph -> P e. ( *Met ` ( X X. Y ) ) )

Distinct variable groups:   u, M, v   u, N, v   u, X, v   u, Y, v

Allowed substitution hints:   ph( v, u)   P( v, u)

Proof of Theorem xmetxp

Dummy variables r s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.1	. . . 4 |- ( ph -> M e. ( *Met ` X ) )
2		eqid 2139	. . . . 5 |- ( MetOpen ` M ) = ( MetOpen ` M )
3	2	mopnm 12617	. . . 4 |- ( M e. ( *Met ` X ) -> X e. ( MetOpen ` M ) )
4	1, 3	syl 14	. . 3 |- ( ph -> X e. ( MetOpen ` M ) )
5		xmetxp.2	. . . 4 |- ( ph -> N e. ( *Met ` Y ) )
6		eqid 2139	. . . . 5 |- ( MetOpen ` N ) = ( MetOpen ` N )
7	6	mopnm 12617	. . . 4 |- ( N e. ( *Met ` Y ) -> Y e. ( MetOpen ` N ) )
8	5, 7	syl 14	. . 3 |- ( ph -> Y e. ( MetOpen ` N ) )
9		xpexg 4653	. . 3 |- ( ( X e. ( MetOpen ` M ) /\ Y e. ( MetOpen ` N ) ) -> ( X X. Y ) e. _V )
10	4, 8, 9	syl2anc 408	. 2 |- ( ph -> ( X X. Y ) e. _V )
11	1	adantr 274	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> M e. ( *Met ` X ) )
12		xp1st 6063	. . . . . . 7 |- ( r e. ( X X. Y ) -> ( 1st ` r ) e. X )
13	12	ad2antrl 481	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 1st ` r ) e. X )
14		xp1st 6063	. . . . . . 7 |- ( s e. ( X X. Y ) -> ( 1st ` s ) e. X )
15	14	ad2antll 482	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 1st ` s ) e. X )
16		xmetcl 12521	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` r ) e. X /\ ( 1st ` s ) e. X ) -> ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* )
17	11, 13, 15, 16	syl3anc 1216	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* )
18	5	adantr 274	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> N e. ( *Met ` Y ) )
19		xp2nd 6064	. . . . . . 7 |- ( r e. ( X X. Y ) -> ( 2nd ` r ) e. Y )
20	19	ad2antrl 481	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 2nd ` r ) e. Y )
21		xp2nd 6064	. . . . . . 7 |- ( s e. ( X X. Y ) -> ( 2nd ` s ) e. Y )
22	21	ad2antll 482	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 2nd ` s ) e. Y )
23		xmetcl 12521	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` r ) e. Y /\ ( 2nd ` s ) e. Y ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* )
24	18, 20, 22, 23	syl3anc 1216	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* )
25		xrmaxcl 11021	. . . . 5 |- ( ( ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* ) -> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* )
26	17, 24, 25	syl2anc 408	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* )
27	26	ralrimivva 2514	. . 3 |- ( ph -> A. r e. ( X X. Y ) A. s e. ( X X. Y ) sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* )
28		xmetxp.p	. . . . 5 |- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
29		fveq2 5421	. . . . . . . . 9 |- ( u = r -> ( 1st ` u ) = ( 1st ` r ) )
30	29	oveq1d 5789	. . . . . . . 8 |- ( u = r -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` v ) ) )
31		fveq2 5421	. . . . . . . . 9 |- ( u = r -> ( 2nd ` u ) = ( 2nd ` r ) )
32	31	oveq1d 5789	. . . . . . . 8 |- ( u = r -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` v ) ) )
33	30, 32	preq12d 3608	. . . . . . 7 |- ( u = r -> { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } = { ( ( 1st ` r ) M ( 1st ` v ) ) , ( ( 2nd ` r ) N ( 2nd ` v ) ) } )
34	33	supeq1d 6874	. . . . . 6 |- ( u = r -> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) = sup ( { ( ( 1st ` r ) M ( 1st ` v ) ) , ( ( 2nd ` r ) N ( 2nd ` v ) ) } , RR* , < ) )
35		fveq2 5421	. . . . . . . . 9 |- ( v = s -> ( 1st ` v ) = ( 1st ` s ) )
36	35	oveq2d 5790	. . . . . . . 8 |- ( v = s -> ( ( 1st ` r ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` s ) ) )
37		fveq2 5421	. . . . . . . . 9 |- ( v = s -> ( 2nd ` v ) = ( 2nd ` s ) )
38	37	oveq2d 5790	. . . . . . . 8 |- ( v = s -> ( ( 2nd ` r ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` s ) ) )
39	36, 38	preq12d 3608	. . . . . . 7 |- ( v = s -> { ( ( 1st ` r ) M ( 1st ` v ) ) , ( ( 2nd ` r ) N ( 2nd ` v ) ) } = { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } )
40	39	supeq1d 6874	. . . . . 6 |- ( v = s -> sup ( { ( ( 1st ` r ) M ( 1st ` v ) ) , ( ( 2nd ` r ) N ( 2nd ` v ) ) } , RR* , < ) = sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
41	34, 40	cbvmpov 5851	. . . . 5 |- ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) ) = ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
42	28, 41	eqtri 2160	. . . 4 |- P = ( r e. ( X X. Y ) , s e. ( X X. Y ) |-> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
43	42	fmpo 6099	. . 3 |- ( A. r e. ( X X. Y ) A. s e. ( X X. Y ) sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* <-> P : ( ( X X. Y ) X. ( X X. Y ) ) --> RR* )
44	27, 43	sylib 121	. 2 |- ( ph -> P : ( ( X X. Y ) X. ( X X. Y ) ) --> RR* )
45		simprl 520	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> r e. ( X X. Y ) )
46		simprr 521	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> s e. ( X X. Y ) )
47	34, 40, 28	ovmpog 5905	. . . . . . . 8 |- ( ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* ) -> ( r P s ) = sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
48	45, 46, 26, 47	syl3anc 1216	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( r P s ) = sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
49	48, 26	eqeltrd 2216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( r P s ) e. RR* )
50		0xr 7812	. . . . . . 7 |- 0 e. RR*
51	50	a1i 9	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 e. RR* )
52		xrletri3 9588	. . . . . 6 |- ( ( ( r P s ) e. RR* /\ 0 e. RR* ) -> ( ( r P s ) = 0 <-> ( ( r P s ) <_ 0 /\ 0 <_ ( r P s ) ) ) )
53	49, 51, 52	syl2anc 408	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> ( ( r P s ) <_ 0 /\ 0 <_ ( r P s ) ) ) )
54		xmetge0 12534	. . . . . . . . 9 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` r ) e. X /\ ( 1st ` s ) e. X ) -> 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) )
55	11, 13, 15, 54	syl3anc 1216	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) )
56		xrmax1sup 11022	. . . . . . . . 9 |- ( ( ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* ) -> ( ( 1st ` r ) M ( 1st ` s ) ) <_ sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
57	17, 24, 56	syl2anc 408	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) <_ sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
58	51, 17, 26, 55, 57	xrletrd 9595	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
59	58, 48	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( r P s ) )
60	59	biantrud 302	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) <_ 0 <-> ( ( r P s ) <_ 0 /\ 0 <_ ( r P s ) ) ) )
61	53, 60	bitr4d 190	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> ( r P s ) <_ 0 ) )
62	48	breq1d 3939	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) <_ 0 <-> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ 0 ) )
63		xrmaxlesup 11028	. . . . 5 |- ( ( ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* /\ 0 e. RR* ) -> ( sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 ) ) )
64	17, 24, 51, 63	syl3anc 1216	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 ) ) )
65	61, 62, 64	3bitrd 213	. . 3 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 ) ) )
66	55	biantrud 302	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) ) ) )
67		xrletri3 9588	. . . . . 6 |- ( ( ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* /\ 0 e. RR* ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) ) ) )
68	17, 51, 67	syl2anc 408	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) ) ) )
69	66, 68	bitr4d 190	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 <-> ( ( 1st ` r ) M ( 1st ` s ) ) = 0 ) )
70		xmetge0 12534	. . . . . . 7 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` r ) e. Y /\ ( 2nd ` s ) e. Y ) -> 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) )
71	18, 20, 22, 70	syl3anc 1216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) )
72	71	biantrud 302	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 <-> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 /\ 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) ) ) )
73		xrletri3 9588	. . . . . 6 |- ( ( ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* /\ 0 e. RR* ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 <-> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 /\ 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) ) ) )
74	24, 51, 73	syl2anc 408	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 <-> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 /\ 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) ) ) )
75	72, 74	bitr4d 190	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 <-> ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 ) )
76	69, 75	anbi12d 464	. . 3 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ 0 ) <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 ) ) )
77		xmeteq0 12528	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` r ) e. X /\ ( 1st ` s ) e. X ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( 1st ` r ) = ( 1st ` s ) ) )
78	11, 13, 15, 77	syl3anc 1216	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( 1st ` r ) = ( 1st ` s ) ) )
79		xmeteq0 12528	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` r ) e. Y /\ ( 2nd ` s ) e. Y ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 <-> ( 2nd ` r ) = ( 2nd ` s ) ) )
80	18, 20, 22, 79	syl3anc 1216	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 <-> ( 2nd ` r ) = ( 2nd ` s ) ) )
81	78, 80	anbi12d 464	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 ) <-> ( ( 1st ` r ) = ( 1st ` s ) /\ ( 2nd ` r ) = ( 2nd ` s ) ) ) )
82		xpopth 6074	. . . . 5 |- ( ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) -> ( ( ( 1st ` r ) = ( 1st ` s ) /\ ( 2nd ` r ) = ( 2nd ` s ) ) <-> r = s ) )
83	82	adantl 275	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) = ( 1st ` s ) /\ ( 2nd ` r ) = ( 2nd ` s ) ) <-> r = s ) )
84	81, 83	bitrd 187	. . 3 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) = 0 ) <-> r = s ) )
85	65, 76, 84	3bitrd 213	. 2 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> r = s ) )
86	48	3adantr3 1142	. . 3 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( r P s ) = sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) )
87	17	3adantr3 1142	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* )
88	1	adantr 274	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> M e. ( *Met ` X ) )
89		simpr3 989	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> t e. ( X X. Y ) )
90		xp1st 6063	. . . . . . . 8 |- ( t e. ( X X. Y ) -> ( 1st ` t ) e. X )
91	89, 90	syl 14	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 1st ` t ) e. X )
92		simpr1 987	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> r e. ( X X. Y ) )
93	92, 12	syl 14	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 1st ` r ) e. X )
94		xmetcl 12521	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` t ) e. X /\ ( 1st ` r ) e. X ) -> ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* )
95	88, 91, 93, 94	syl3anc 1216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* )
96	15	3adantr3 1142	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 1st ` s ) e. X )
97		xmetcl 12521	. . . . . . 7 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` t ) e. X /\ ( 1st ` s ) e. X ) -> ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* )
98	88, 91, 96, 97	syl3anc 1216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* )
99	95, 98	xaddcld 9667	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) e. RR* )
100	5	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> N e. ( *Met ` Y ) )
101		xp2nd 6064	. . . . . . . . . . 11 |- ( t e. ( X X. Y ) -> ( 2nd ` t ) e. Y )
102	89, 101	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 2nd ` t ) e. Y )
103	92, 19	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 2nd ` r ) e. Y )
104		xmetcl 12521	. . . . . . . . . 10 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` t ) e. Y /\ ( 2nd ` r ) e. Y ) -> ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* )
105	100, 102, 103, 104	syl3anc 1216	. . . . . . . . 9 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* )
106		xrmaxcl 11021	. . . . . . . . 9 |- ( ( ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* ) -> sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) e. RR* )
107	95, 105, 106	syl2anc 408	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) e. RR* )
108		fveq2 5421	. . . . . . . . . . . 12 |- ( u = t -> ( 1st ` u ) = ( 1st ` t ) )
109		fveq2 5421	. . . . . . . . . . . 12 |- ( v = r -> ( 1st ` v ) = ( 1st ` r ) )
110	108, 109	oveqan12d 5793	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` r ) ) )
111		fveq2 5421	. . . . . . . . . . . 12 |- ( u = t -> ( 2nd ` u ) = ( 2nd ` t ) )
112		fveq2 5421	. . . . . . . . . . . 12 |- ( v = r -> ( 2nd ` v ) = ( 2nd ` r ) )
113	111, 112	oveqan12d 5793	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` r ) ) )
114	110, 113	preq12d 3608	. . . . . . . . . 10 |- ( ( u = t /\ v = r ) -> { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } = { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } )
115	114	supeq1d 6874	. . . . . . . . 9 |- ( ( u = t /\ v = r ) -> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) = sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
116	115, 28	ovmpoga 5900	. . . . . . . 8 |- ( ( t e. ( X X. Y ) /\ r e. ( X X. Y ) /\ sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) e. RR* ) -> ( t P r ) = sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
117	89, 92, 107, 116	syl3anc 1216	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P r ) = sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
118	117, 107	eqeltrd 2216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P r ) e. RR* )
119		simpr2 988	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> s e. ( X X. Y ) )
120	22	3adantr3 1142	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 2nd ` s ) e. Y )
121		xmetcl 12521	. . . . . . . . . 10 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` t ) e. Y /\ ( 2nd ` s ) e. Y ) -> ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* )
122	100, 102, 120, 121	syl3anc 1216	. . . . . . . . 9 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* )
123		xrmaxcl 11021	. . . . . . . . 9 |- ( ( ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* ) -> sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* )
124	98, 122, 123	syl2anc 408	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* )
125	108, 35	oveqan12d 5793	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` s ) ) )
126	111, 37	oveqan12d 5793	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` s ) ) )
127	125, 126	preq12d 3608	. . . . . . . . . 10 |- ( ( u = t /\ v = s ) -> { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } = { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } )
128	127	supeq1d 6874	. . . . . . . . 9 |- ( ( u = t /\ v = s ) -> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) = sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
129	128, 28	ovmpoga 5900	. . . . . . . 8 |- ( ( t e. ( X X. Y ) /\ s e. ( X X. Y ) /\ sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) e. RR* ) -> ( t P s ) = sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
130	89, 119, 124, 129	syl3anc 1216	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P s ) = sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
131	130, 124	eqeltrd 2216	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P s ) e. RR* )
132	118, 131	xaddcld 9667	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( t P r ) +e ( t P s ) ) e. RR* )
133		xmettri2 12530	. . . . . 6 |- ( ( M e. ( *Met ` X ) /\ ( ( 1st ` t ) e. X /\ ( 1st ` r ) e. X /\ ( 1st ` s ) e. X ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) <_ ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) )
134	88, 91, 93, 96, 133	syl13anc 1218	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) <_ ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) )
135		xrmax1sup 11022	. . . . . . . 8 |- ( ( ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* ) -> ( ( 1st ` t ) M ( 1st ` r ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
136	95, 105, 135	syl2anc 408	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` r ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
137	136, 117	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` r ) ) <_ ( t P r ) )
138		xrmax1sup 11022	. . . . . . . 8 |- ( ( ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* ) -> ( ( 1st ` t ) M ( 1st ` s ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
139	98, 122, 138	syl2anc 408	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` s ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
140	139, 130	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` t ) M ( 1st ` s ) ) <_ ( t P s ) )
141		xle2add 9662	. . . . . . 7 |- ( ( ( ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* /\ ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* ) /\ ( ( t P r ) e. RR* /\ ( t P s ) e. RR* ) ) -> ( ( ( ( 1st ` t ) M ( 1st ` r ) ) <_ ( t P r ) /\ ( ( 1st ` t ) M ( 1st ` s ) ) <_ ( t P s ) ) -> ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) ) )
142	95, 98, 118, 131, 141	syl22anc 1217	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( ( 1st ` t ) M ( 1st ` r ) ) <_ ( t P r ) /\ ( ( 1st ` t ) M ( 1st ` s ) ) <_ ( t P s ) ) -> ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) ) )
143	137, 140, 142	mp2and 429	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( 1st ` t ) M ( 1st ` r ) ) +e ( ( 1st ` t ) M ( 1st ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) )
144	87, 99, 132, 134, 143	xrletrd 9595	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) )
145	24	3adantr3 1142	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* )
146	105, 122	xaddcld 9667	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) e. RR* )
147		xmettri2 12530	. . . . . 6 |- ( ( N e. ( *Met ` Y ) /\ ( ( 2nd ` t ) e. Y /\ ( 2nd ` r ) e. Y /\ ( 2nd ` s ) e. Y ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) )
148	100, 102, 103, 120, 147	syl13anc 1218	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) )
149		xrmax2sup 11023	. . . . . . . 8 |- ( ( ( ( 1st ` t ) M ( 1st ` r ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* ) -> ( ( 2nd ` t ) N ( 2nd ` r ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
150	95, 105, 149	syl2anc 408	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` r ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` r ) ) , ( ( 2nd ` t ) N ( 2nd ` r ) ) } , RR* , < ) )
151	150, 117	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` r ) ) <_ ( t P r ) )
152		xrmax2sup 11023	. . . . . . . 8 |- ( ( ( ( 1st ` t ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* ) -> ( ( 2nd ` t ) N ( 2nd ` s ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
153	98, 122, 152	syl2anc 408	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` s ) ) <_ sup ( { ( ( 1st ` t ) M ( 1st ` s ) ) , ( ( 2nd ` t ) N ( 2nd ` s ) ) } , RR* , < ) )
154	153, 130	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` t ) N ( 2nd ` s ) ) <_ ( t P s ) )
155		xle2add 9662	. . . . . . 7 |- ( ( ( ( ( 2nd ` t ) N ( 2nd ` r ) ) e. RR* /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) e. RR* ) /\ ( ( t P r ) e. RR* /\ ( t P s ) e. RR* ) ) -> ( ( ( ( 2nd ` t ) N ( 2nd ` r ) ) <_ ( t P r ) /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) <_ ( t P s ) ) -> ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) ) )
156	105, 122, 118, 131, 155	syl22anc 1217	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( ( 2nd ` t ) N ( 2nd ` r ) ) <_ ( t P r ) /\ ( ( 2nd ` t ) N ( 2nd ` s ) ) <_ ( t P s ) ) -> ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) ) )
157	151, 154, 156	mp2and 429	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( ( 2nd ` t ) N ( 2nd ` r ) ) +e ( ( 2nd ` t ) N ( 2nd ` s ) ) ) <_ ( ( t P r ) +e ( t P s ) ) )
158	145, 146, 132, 148, 157	xrletrd 9595	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) )
159		xrmaxlesup 11028	. . . . 5 |- ( ( ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* /\ ( ( t P r ) +e ( t P s ) ) e. RR* ) -> ( sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ ( ( t P r ) +e ( t P s ) ) <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) ) ) )
160	87, 145, 132, 159	syl3anc 1216	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ ( ( t P r ) +e ( t P s ) ) <-> ( ( ( 1st ` r ) M ( 1st ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) /\ ( ( 2nd ` r ) N ( 2nd ` s ) ) <_ ( ( t P r ) +e ( t P s ) ) ) ) )
161	144, 158, 160	mpbir2and 928	. . 3 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> sup ( { ( ( 1st ` r ) M ( 1st ` s ) ) , ( ( 2nd ` r ) N ( 2nd ` s ) ) } , RR* , < ) <_ ( ( t P r ) +e ( t P s ) ) )
162	86, 161	eqbrtrd 3950	. 2 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( r P s ) <_ ( ( t P r ) +e ( t P s ) ) )
163	10, 44, 85, 162	isxmetd 12516	1 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   _Vcvv 2686   {cpr 3528   class class class wbr 3929   X. cxp 4537   -->wf 5119   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   supcsup 6869   0cc0 7620   RR*cxr 7799   < clt 7800   <_ cle 7801   +ecxad 9557   *Metcxmet 12149   MetOpencmopn 12154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210

This theorem is referenced by:  xmetxpbl  12677  xmettxlem  12678  xmettx  12679  txmetcnp  12687