Theorem tmsxpsval 22343

Description: Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tmsxps.p	|- P = ( dist ` ( (toMetSp ` M ) X.s (toMetSp ` N ) ) )
tmsxps.1	|- ( ph -> M e. ( *Met ` X ) )
tmsxps.2	|- ( ph -> N e. ( *Met ` Y ) )
tmsxpsval.a	|- ( ph -> A e. X )
tmsxpsval.b	|- ( ph -> B e. Y )
tmsxpsval.c	|- ( ph -> C e. X )
tmsxpsval.d	|- ( ph -> D e. Y )

Assertion

Ref	Expression
tmsxpsval	|- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Proof of Theorem tmsxpsval

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( (toMetSp ` M ) X.s (toMetSp ` N ) ) = ( (toMetSp ` M ) X.s (toMetSp ` N ) )
2		eqid 2622	. . 3 |- ( Base ` (toMetSp ` M ) ) = ( Base ` (toMetSp ` M ) )
3		eqid 2622	. . 3 |- ( Base ` (toMetSp ` N ) ) = ( Base ` (toMetSp ` N ) )
4		tmsxps.1	. . . 4 |- ( ph -> M e. ( *Met ` X ) )
5		eqid 2622	. . . . 5 |- (toMetSp ` M ) = (toMetSp ` M )
6	5	tmsxms 22291	. . . 4 |- ( M e. ( *Met ` X ) -> (toMetSp ` M ) e. *MetSp )
7	4, 6	syl 17	. . 3 |- ( ph -> (toMetSp ` M ) e. *MetSp )
8		tmsxps.2	. . . 4 |- ( ph -> N e. ( *Met ` Y ) )
9		eqid 2622	. . . . 5 |- (toMetSp ` N ) = (toMetSp ` N )
10	9	tmsxms 22291	. . . 4 |- ( N e. ( *Met ` Y ) -> (toMetSp ` N ) e. *MetSp )
11	8, 10	syl 17	. . 3 |- ( ph -> (toMetSp ` N ) e. *MetSp )
12		tmsxps.p	. . 3 |- P = ( dist ` ( (toMetSp ` M ) X.s (toMetSp ` N ) ) )
13		eqid 2622	. . 3 |- ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) = ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) )
14		eqid 2622	. . 3 |- ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) = ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) )
15	5	tmsds 22289	. . . . . 6 |- ( M e. ( *Met ` X ) -> M = ( dist ` (toMetSp ` M ) ) )
16	4, 15	syl 17	. . . . 5 |- ( ph -> M = ( dist ` (toMetSp ` M ) ) )
17	5	tmsbas 22288	. . . . . . 7 |- ( M e. ( *Met ` X ) -> X = ( Base ` (toMetSp ` M ) ) )
18	4, 17	syl 17	. . . . . 6 |- ( ph -> X = ( Base ` (toMetSp ` M ) ) )
19	18	fveq2d 6195	. . . . 5 |- ( ph -> ( *Met ` X ) = ( *Met ` ( Base ` (toMetSp ` M ) ) ) )
20	4, 16, 19	3eltr3d 2715	. . . 4 |- ( ph -> ( dist ` (toMetSp ` M ) ) e. ( *Met ` ( Base ` (toMetSp ` M ) ) ) )
21		ssid 3624	. . . 4 |- ( Base ` (toMetSp ` M ) ) C_ ( Base ` (toMetSp ` M ) )
22		xmetres2 22166	. . . 4 |- ( ( ( dist ` (toMetSp ` M ) ) e. ( *Met ` ( Base ` (toMetSp ` M ) ) ) /\ ( Base ` (toMetSp ` M ) ) C_ ( Base ` (toMetSp ` M ) ) ) -> ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) e. ( *Met ` ( Base ` (toMetSp ` M ) ) ) )
23	20, 21, 22	sylancl 694	. . 3 |- ( ph -> ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) e. ( *Met ` ( Base ` (toMetSp ` M ) ) ) )
24	9	tmsds 22289	. . . . . 6 |- ( N e. ( *Met ` Y ) -> N = ( dist ` (toMetSp ` N ) ) )
25	8, 24	syl 17	. . . . 5 |- ( ph -> N = ( dist ` (toMetSp ` N ) ) )
26	9	tmsbas 22288	. . . . . . 7 |- ( N e. ( *Met ` Y ) -> Y = ( Base ` (toMetSp ` N ) ) )
27	8, 26	syl 17	. . . . . 6 |- ( ph -> Y = ( Base ` (toMetSp ` N ) ) )
28	27	fveq2d 6195	. . . . 5 |- ( ph -> ( *Met ` Y ) = ( *Met ` ( Base ` (toMetSp ` N ) ) ) )
29	8, 25, 28	3eltr3d 2715	. . . 4 |- ( ph -> ( dist ` (toMetSp ` N ) ) e. ( *Met ` ( Base ` (toMetSp ` N ) ) ) )
30		ssid 3624	. . . 4 |- ( Base ` (toMetSp ` N ) ) C_ ( Base ` (toMetSp ` N ) )
31		xmetres2 22166	. . . 4 |- ( ( ( dist ` (toMetSp ` N ) ) e. ( *Met ` ( Base ` (toMetSp ` N ) ) ) /\ ( Base ` (toMetSp ` N ) ) C_ ( Base ` (toMetSp ` N ) ) ) -> ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) e. ( *Met ` ( Base ` (toMetSp ` N ) ) ) )
32	29, 30, 31	sylancl 694	. . 3 |- ( ph -> ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) e. ( *Met ` ( Base ` (toMetSp ` N ) ) ) )
33		tmsxpsval.a	. . . 4 |- ( ph -> A e. X )
34	33, 18	eleqtrd 2703	. . 3 |- ( ph -> A e. ( Base ` (toMetSp ` M ) ) )
35		tmsxpsval.b	. . . 4 |- ( ph -> B e. Y )
36	35, 27	eleqtrd 2703	. . 3 |- ( ph -> B e. ( Base ` (toMetSp ` N ) ) )
37		tmsxpsval.c	. . . 4 |- ( ph -> C e. X )
38	37, 18	eleqtrd 2703	. . 3 |- ( ph -> C e. ( Base ` (toMetSp ` M ) ) )
39		tmsxpsval.d	. . . 4 |- ( ph -> D e. Y )
40	39, 27	eleqtrd 2703	. . 3 |- ( ph -> D e. ( Base ` (toMetSp ` N ) ) )
41	1, 2, 3, 7, 11, 12, 13, 14, 23, 32, 34, 36, 38, 40	xpsdsval 22186	. 2 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) C ) , ( B ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) D ) } , RR* , < ) )
42	34, 38	ovresd 6801	. . . . 5 |- ( ph -> ( A ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) C ) = ( A ( dist ` (toMetSp ` M ) ) C ) )
43	16	oveqd 6667	. . . . 5 |- ( ph -> ( A M C ) = ( A ( dist ` (toMetSp ` M ) ) C ) )
44	42, 43	eqtr4d 2659	. . . 4 |- ( ph -> ( A ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) C ) = ( A M C ) )
45	36, 40	ovresd 6801	. . . . 5 |- ( ph -> ( B ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) D ) = ( B ( dist ` (toMetSp ` N ) ) D ) )
46	25	oveqd 6667	. . . . 5 |- ( ph -> ( B N D ) = ( B ( dist ` (toMetSp ` N ) ) D ) )
47	45, 46	eqtr4d 2659	. . . 4 |- ( ph -> ( B ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) D ) = ( B N D ) )
48	44, 47	preq12d 4276	. . 3 |- ( ph -> { ( A ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) C ) , ( B ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) D ) } = { ( A M C ) , ( B N D ) } )
49	48	supeq1d 8352	. 2 |- ( ph -> sup ( { ( A ( ( dist ` (toMetSp ` M ) ) |` ( ( Base ` (toMetSp ` M ) ) X. ( Base ` (toMetSp ` M ) ) ) ) C ) , ( B ( ( dist ` (toMetSp ` N ) ) |` ( ( Base ` (toMetSp ` N ) ) X. ( Base ` (toMetSp ` N ) ) ) ) D ) } , RR* , < ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
50	41, 49	eqtrd 2656	1 |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   {cpr 4179   <.cop 4183   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   supcsup 8346   RR*cxr 10073   < clt 10074   Basecbs 15857   distcds 15950   X._s cxps 16166   *Metcxmt 19731   *MetSpcxme 22122  toMetSpctmt 22124

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-prds 16108  df-xrs 16162  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-tms 22127

This theorem is referenced by:  tmsxpsval2  22344