Theorem xmetxp 14390

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 2188	. . . . 5 |- ( MetOpen ` M ) = ( MetOpen ` M )
3	2	mopnm 14331	. . . 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 2188	. . . . 5 |- ( MetOpen ` N ) = ( MetOpen ` N )
7	6	mopnm 14331	. . . 4 |- ( N e. ( *Met ` Y ) -> Y e. ( MetOpen ` N ) )
8	5, 7	syl 14	. . 3 |- ( ph -> Y e. ( MetOpen ` N ) )
9		xpexg 4754	. . 3 |- ( ( X e. ( MetOpen ` M ) /\ Y e. ( MetOpen ` N ) ) -> ( X X. Y ) e. _V )
10	4, 8, 9	syl2anc 411	. 2 |- ( ph -> ( X X. Y ) e. _V )
11	1	adantr 276	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> M e. ( *Met ` X ) )
12		xp1st 6183	. . . . . . 7 |- ( r e. ( X X. Y ) -> ( 1st ` r ) e. X )
13	12	ad2antrl 490	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 1st ` r ) e. X )
14		xp1st 6183	. . . . . . 7 |- ( s e. ( X X. Y ) -> ( 1st ` s ) e. X )
15	14	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 1st ` s ) e. X )
16		xmetcl 14235	. . . . . 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 1248	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 1st ` r ) M ( 1st ` s ) ) e. RR* )
18	5	adantr 276	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> N e. ( *Met ` Y ) )
19		xp2nd 6184	. . . . . . 7 |- ( r e. ( X X. Y ) -> ( 2nd ` r ) e. Y )
20	19	ad2antrl 490	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 2nd ` r ) e. Y )
21		xp2nd 6184	. . . . . . 7 |- ( s e. ( X X. Y ) -> ( 2nd ` s ) e. Y )
22	21	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( 2nd ` s ) e. Y )
23		xmetcl 14235	. . . . . 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 1248	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* )
25		xrmaxcl 11277	. . . . 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 411	. . . 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 2571	. . 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 5529	. . . . . . . . 9 |- ( u = r -> ( 1st ` u ) = ( 1st ` r ) )
30	29	oveq1d 5905	. . . . . . . 8 |- ( u = r -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` v ) ) )
31		fveq2 5529	. . . . . . . . 9 |- ( u = r -> ( 2nd ` u ) = ( 2nd ` r ) )
32	31	oveq1d 5905	. . . . . . . 8 |- ( u = r -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` v ) ) )
33	30, 32	preq12d 3691	. . . . . . 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 7003	. . . . . 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 5529	. . . . . . . . 9 |- ( v = s -> ( 1st ` v ) = ( 1st ` s ) )
36	35	oveq2d 5906	. . . . . . . 8 |- ( v = s -> ( ( 1st ` r ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` s ) ) )
37		fveq2 5529	. . . . . . . . 9 |- ( v = s -> ( 2nd ` v ) = ( 2nd ` s ) )
38	37	oveq2d 5906	. . . . . . . 8 |- ( v = s -> ( ( 2nd ` r ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` s ) ) )
39	36, 38	preq12d 3691	. . . . . . 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 7003	. . . . . 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 5970	. . . . 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 2209	. . . 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 6219	. . 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 122	. 2 |- ( ph -> P : ( ( X X. Y ) X. ( X X. Y ) ) --> RR* )
45		simprl 529	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> r e. ( X X. Y ) )
46		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> s e. ( X X. Y ) )
47	34, 40, 28	ovmpog 6025	. . . . . . . 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 1248	. . . . . . 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 2265	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( r P s ) e. RR* )
50		0xr 8021	. . . . . . 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 9821	. . . . . 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 411	. . . . 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 14248	. . . . . . . . 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 1248	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) )
56		xrmax1sup 11278	. . . . . . . . 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 411	. . . . . . . 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 9829	. . . . . . 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 4045	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( r P s ) )
60	59	biantrud 304	. . . . 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 191	. . . 4 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> ( r P s ) <_ 0 ) )
62	48	breq1d 4027	. . . 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 11284	. . . . 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 1248	. . . 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 214	. . 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 304	. . . . 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 9821	. . . . . 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 411	. . . . 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 191	. . . 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 14248	. . . . . . 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 1248	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( ( 2nd ` r ) N ( 2nd ` s ) ) )
72	71	biantrud 304	. . . . 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 9821	. . . . . 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 411	. . . . 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 191	. . . 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 473	. . 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 14242	. . . . . 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 1248	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( 1st ` r ) = ( 1st ` s ) ) )
79		xmeteq0 14242	. . . . . 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 1248	. . . . 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 473	. . . 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 6194	. . . . 5 |- ( ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) -> ( ( ( 1st ` r ) = ( 1st ` s ) /\ ( 2nd ` r ) = ( 2nd ` s ) ) <-> r = s ) )
83	82	adantl 277	. . . 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 188	. . 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 214	. 2 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( r P s ) = 0 <-> r = s ) )
86	48	3adantr3 1159	. . 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 1159	. . . . 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 276	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> M e. ( *Met ` X ) )
89		simpr3 1006	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> t e. ( X X. Y ) )
90		xp1st 6183	. . . . . . . 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 1004	. . . . . . . 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 14235	. . . . . . 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 1248	. . . . . 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 1159	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 1st ` s ) e. X )
97		xmetcl 14235	. . . . . . 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 1248	. . . . . 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 9901	. . . . 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 276	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> N e. ( *Met ` Y ) )
101		xp2nd 6184	. . . . . . . . . . 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 14235	. . . . . . . . . 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 1248	. . . . . . . . 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 11277	. . . . . . . . 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 411	. . . . . . . 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 5529	. . . . . . . . . . . 12 |- ( u = t -> ( 1st ` u ) = ( 1st ` t ) )
109		fveq2 5529	. . . . . . . . . . . 12 |- ( v = r -> ( 1st ` v ) = ( 1st ` r ) )
110	108, 109	oveqan12d 5909	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` r ) ) )
111		fveq2 5529	. . . . . . . . . . . 12 |- ( u = t -> ( 2nd ` u ) = ( 2nd ` t ) )
112		fveq2 5529	. . . . . . . . . . . 12 |- ( v = r -> ( 2nd ` v ) = ( 2nd ` r ) )
113	111, 112	oveqan12d 5909	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` r ) ) )
114	110, 113	preq12d 3691	. . . . . . . . . 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 7003	. . . . . . . . 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 6020	. . . . . . . 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 1248	. . . . . . 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 2265	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P r ) e. RR* )
119		simpr2 1005	. . . . . . . 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 1159	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 2nd ` s ) e. Y )
121		xmetcl 14235	. . . . . . . . . 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 1248	. . . . . . . . 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 11277	. . . . . . . . 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 411	. . . . . . . 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 5909	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` s ) ) )
126	111, 37	oveqan12d 5909	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` s ) ) )
127	125, 126	preq12d 3691	. . . . . . . . . 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 7003	. . . . . . . . 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 6020	. . . . . . . 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 1248	. . . . . . 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 2265	. . . . . 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 9901	. . . . 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 14244	. . . . . 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 1250	. . . . 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 11278	. . . . . . . 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 411	. . . . . . 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 4045	. . . . . 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 11278	. . . . . . . 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 411	. . . . . . 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 4045	. . . . . 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 9896	. . . . . . 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 1249	. . . . . 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 433	. . . . 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 9829	. . . 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 1159	. . . . 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 9901	. . . . 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 14244	. . . . . 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 1250	. . . . 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 11279	. . . . . . . 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 411	. . . . . . 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 4045	. . . . . 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 11279	. . . . . . . 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 411	. . . . . . 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 4045	. . . . . 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 9896	. . . . . . 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 1249	. . . . . 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 433	. . . . 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 9829	. . . 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 11284	. . . . 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 1248	. . . 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 945	. . 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 4039	. 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 14230	1 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2159   A.wral 2467   _Vcvv 2751   {cpr 3607   class class class wbr 4017   X. cxp 4638   -->wf 5226   ` cfv 5230  (class class class)co 5890   e. cmpo 5892   1stc1st 6156   2ndc2nd 6157   supcsup 6998   0cc0 7828   RR*cxr 8008   < clt 8009   <_ cle 8010   +ecxad 9787   *Metcxmet 13809   MetOpencmopn 13814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-map 6667  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-xneg 9789  df-xadd 9790  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-topgen 12730  df-psmet 13816  df-xmet 13817  df-bl 13819  df-mopn 13820  df-top 13881  df-topon 13894  df-bases 13926

This theorem is referenced by:  xmetxpbl  14391  xmettxlem  14392  xmettx  14393  txmetcnp  14401