Theorem xmetxp 13977

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 2177	. . . . 5 |- ( MetOpen ` M ) = ( MetOpen ` M )
3	2	mopnm 13918	. . . 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 2177	. . . . 5 |- ( MetOpen ` N ) = ( MetOpen ` N )
7	6	mopnm 13918	. . . 4 |- ( N e. ( *Met ` Y ) -> Y e. ( MetOpen ` N ) )
8	5, 7	syl 14	. . 3 |- ( ph -> Y e. ( MetOpen ` N ) )
9		xpexg 4740	. . 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 6165	. . . . . . 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 6165	. . . . . . 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 13822	. . . . . 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 1238	. . . . 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 6166	. . . . . . 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 6166	. . . . . . 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 13822	. . . . . 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 1238	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( 2nd ` r ) N ( 2nd ` s ) ) e. RR* )
25		xrmaxcl 11259	. . . . 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 2559	. . 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 5515	. . . . . . . . 9 |- ( u = r -> ( 1st ` u ) = ( 1st ` r ) )
30	29	oveq1d 5889	. . . . . . . 8 |- ( u = r -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` v ) ) )
31		fveq2 5515	. . . . . . . . 9 |- ( u = r -> ( 2nd ` u ) = ( 2nd ` r ) )
32	31	oveq1d 5889	. . . . . . . 8 |- ( u = r -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` v ) ) )
33	30, 32	preq12d 3677	. . . . . . 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 6985	. . . . . 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 5515	. . . . . . . . 9 |- ( v = s -> ( 1st ` v ) = ( 1st ` s ) )
36	35	oveq2d 5890	. . . . . . . 8 |- ( v = s -> ( ( 1st ` r ) M ( 1st ` v ) ) = ( ( 1st ` r ) M ( 1st ` s ) ) )
37		fveq2 5515	. . . . . . . . 9 |- ( v = s -> ( 2nd ` v ) = ( 2nd ` s ) )
38	37	oveq2d 5890	. . . . . . . 8 |- ( v = s -> ( ( 2nd ` r ) N ( 2nd ` v ) ) = ( ( 2nd ` r ) N ( 2nd ` s ) ) )
39	36, 38	preq12d 3677	. . . . . . 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 6985	. . . . . 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 5954	. . . . 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 2198	. . . 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 6201	. . 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 6008	. . . . . . . 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 1238	. . . . . . 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 2254	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( r P s ) e. RR* )
50		0xr 8003	. . . . . . 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 9803	. . . . . 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 13835	. . . . . . . . 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 1238	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> 0 <_ ( ( 1st ` r ) M ( 1st ` s ) ) )
56		xrmax1sup 11260	. . . . . . . . 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 9811	. . . . . . 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 4031	. . . . . 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 4013	. . . 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 11266	. . . . 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 1238	. . . 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 9803	. . . . . 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 13835	. . . . . . 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 1238	. . . . . 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 9803	. . . . . 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 13829	. . . . . 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 1238	. . . . 5 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) ) ) -> ( ( ( 1st ` r ) M ( 1st ` s ) ) = 0 <-> ( 1st ` r ) = ( 1st ` s ) ) )
79		xmeteq0 13829	. . . . . 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 1238	. . . . 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 6176	. . . . 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 1158	. . 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 1158	. . . . 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 1005	. . . . . . . 8 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> t e. ( X X. Y ) )
90		xp1st 6165	. . . . . . . 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 1003	. . . . . . . 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 13822	. . . . . . 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 1238	. . . . . 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 1158	. . . . . . 7 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 1st ` s ) e. X )
97		xmetcl 13822	. . . . . . 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 1238	. . . . . 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 9883	. . . . 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 6166	. . . . . . . . . . 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 13822	. . . . . . . . . 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 1238	. . . . . . . . 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 11259	. . . . . . . . 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 5515	. . . . . . . . . . . 12 |- ( u = t -> ( 1st ` u ) = ( 1st ` t ) )
109		fveq2 5515	. . . . . . . . . . . 12 |- ( v = r -> ( 1st ` v ) = ( 1st ` r ) )
110	108, 109	oveqan12d 5893	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` r ) ) )
111		fveq2 5515	. . . . . . . . . . . 12 |- ( u = t -> ( 2nd ` u ) = ( 2nd ` t ) )
112		fveq2 5515	. . . . . . . . . . . 12 |- ( v = r -> ( 2nd ` v ) = ( 2nd ` r ) )
113	111, 112	oveqan12d 5893	. . . . . . . . . . 11 |- ( ( u = t /\ v = r ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` r ) ) )
114	110, 113	preq12d 3677	. . . . . . . . . 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 6985	. . . . . . . . 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 6003	. . . . . . . 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 1238	. . . . . . 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 2254	. . . . . 6 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( t P r ) e. RR* )
119		simpr2 1004	. . . . . . . 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 1158	. . . . . . . . . 10 |- ( ( ph /\ ( r e. ( X X. Y ) /\ s e. ( X X. Y ) /\ t e. ( X X. Y ) ) ) -> ( 2nd ` s ) e. Y )
121		xmetcl 13822	. . . . . . . . . 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 1238	. . . . . . . . 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 11259	. . . . . . . . 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 5893	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 1st ` u ) M ( 1st ` v ) ) = ( ( 1st ` t ) M ( 1st ` s ) ) )
126	111, 37	oveqan12d 5893	. . . . . . . . . . 11 |- ( ( u = t /\ v = s ) -> ( ( 2nd ` u ) N ( 2nd ` v ) ) = ( ( 2nd ` t ) N ( 2nd ` s ) ) )
127	125, 126	preq12d 3677	. . . . . . . . . 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 6985	. . . . . . . . 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 6003	. . . . . . . 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 1238	. . . . . . 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 2254	. . . . . 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 9883	. . . . 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 13831	. . . . . 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 1240	. . . . 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 11260	. . . . . . . 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 4031	. . . . . 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 11260	. . . . . . . 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 4031	. . . . . 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 9878	. . . . . . 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 1239	. . . . . 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 9811	. . . 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 1158	. . . . 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 9883	. . . . 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 13831	. . . . . 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 1240	. . . . 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 11261	. . . . . . . 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 4031	. . . . . 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 11261	. . . . . . . 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 4031	. . . . . 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 9878	. . . . . . 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 1239	. . . . . 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 9811	. . . 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 11266	. . . . 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 1238	. . . 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 944	. . 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 4025	. 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 13817	1 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   {cpr 3593   class class class wbr 4003   X. cxp 4624   -->wf 5212   ` cfv 5216  (class class class)co 5874   e. cmpo 5876   1stc1st 6138   2ndc2nd 6139   supcsup 6980   0cc0 7810   RR*cxr 7990   < clt 7991   <_ cle 7992   +ecxad 9769   *Metcxmet 13410   MetOpencmopn 13415

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-xneg 9771  df-xadd 9772  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-topgen 12708  df-psmet 13417  df-xmet 13418  df-bl 13420  df-mopn 13421  df-top 13468  df-topon 13481  df-bases 13513

This theorem is referenced by:  xmetxpbl  13978  xmettxlem  13979  xmettx  13980  txmetcnp  13988