Theorem xpsmet 21328

Description: The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
xpsds.t	|- T = ( R X.s S )
xpsds.x	|- X = ( Base ` R )
xpsds.y	|- Y = ( Base ` S )
xpsds.1	|- ( ph -> R e. V )
xpsds.2	|- ( ph -> S e. W )
xpsds.p	|- P = ( dist ` T )
xpsds.m	|- M = ( ( dist ` R ) |` ( X X. X ) )
xpsds.n	|- N = ( ( dist ` S ) |` ( Y X. Y ) )
xpsmet.3	|- ( ph -> M e. ( Met ` X ) )
xpsmet.4	|- ( ph -> N e. ( Met ` Y ) )

Assertion

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

Proof of Theorem xpsmet

Dummy variables x k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsds.t	. . 3 |- T = ( R X.s S )
2		xpsds.x	. . 3 |- X = ( Base ` R )
3		xpsds.y	. . 3 |- Y = ( Base ` S )
4		xpsds.1	. . 3 |- ( ph -> R e. V )
5		xpsds.2	. . 3 |- ( ph -> S e. W )
6		eqid 2429	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2429	. . 3 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2429	. . 3 |- ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s `' ( { R } +c { S } ) )
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 15429	. 2 |- ( ph -> T = ( `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) "s ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8	xpslem 15430	. 2 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 15428	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
12		f1ocnv 5843	. . 3 |- ( ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -> `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -1-1-onto-> ( X X. Y ) )
13	11, 12	mp1i 13	. 2 |- ( ph -> `' ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) : ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) -1-1-onto-> ( X X. Y ) )
14		ovex 6333	. . 3 |- ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V
15	14	a1i 11	. 2 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V )
16		eqid 2429	. 2 |- ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) = ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
17		xpsds.p	. 2 |- P = ( dist ` T )
18		eqid 2429	. . . . 5 |- ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
19		eqid 2429	. . . . 5 |- ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
20		eqid 2429	. . . . 5 |- ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` ( `' ( { R } +c { S } ) ` k ) )
21		eqid 2429	. . . . 5 |- ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) )
22		eqid 2429	. . . . 5 |- ( dist ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( dist ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
23		fvex 5891	. . . . . 6 |- (Scalar ` R ) e. _V
24	23	a1i 11	. . . . 5 |- ( ph -> (Scalar ` R ) e. _V )
25		2onn 7349	. . . . . 6 |- 2o e. om
26		nnfi 7771	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
27	25, 26	mp1i 13	. . . . 5 |- ( ph -> 2o e. Fin )
28		fvex 5891	. . . . . 6 |- ( `' ( { R } +c { S } ) ` k ) e. _V
29	28	a1i 11	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) e. _V )
30		elpri 4022	. . . . . . 7 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
31		df2o3 7203	. . . . . . 7 |- 2o = { (/) , 1o }
32	30, 31	eleq2s 2537	. . . . . 6 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
33		xpsmet.3	. . . . . . . . 9 |- ( ph -> M e. ( Met ` X ) )
34	33	adantr 466	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> M e. ( Met ` X ) )
35		fveq2 5881	. . . . . . . . . . . 12 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
36		xpsc0 15417	. . . . . . . . . . . . 13 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
37	4, 36	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
38	35, 37	sylan9eqr 2492	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( `' ( { R } +c { S } ) ` k ) = R )
39	38	fveq2d 5885	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` R ) )
40	38	fveq2d 5885	. . . . . . . . . . . 12 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` R ) )
41	40, 2	syl6eqr 2488	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = X )
42	41	sqxpeqd 4880	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( X X. X ) )
43	39, 42	reseq12d 5126	. . . . . . . . 9 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( ( dist ` R ) |` ( X X. X ) ) )
44		xpsds.m	. . . . . . . . 9 |- M = ( ( dist ` R ) |` ( X X. X ) )
45	43, 44	syl6eqr 2488	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = M )
46	41	fveq2d 5885	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` X ) )
47	34, 45, 46	3eltr4d 2532	. . . . . . 7 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) e. ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) )
48		xpsmet.4	. . . . . . . . 9 |- ( ph -> N e. ( Met ` Y ) )
49	48	adantr 466	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> N e. ( Met ` Y ) )
50		fveq2 5881	. . . . . . . . . . . 12 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
51		xpsc1 15418	. . . . . . . . . . . . 13 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
52	5, 51	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
53	50, 52	sylan9eqr 2492	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( `' ( { R } +c { S } ) ` k ) = S )
54	53	fveq2d 5885	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` S ) )
55	53	fveq2d 5885	. . . . . . . . . . . 12 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` S ) )
56	55, 3	syl6eqr 2488	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = Y )
57	56	sqxpeqd 4880	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Y X. Y ) )
58	54, 57	reseq12d 5126	. . . . . . . . 9 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( ( dist ` S ) |` ( Y X. Y ) ) )
59		xpsds.n	. . . . . . . . 9 |- N = ( ( dist ` S ) |` ( Y X. Y ) )
60	58, 59	syl6eqr 2488	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = N )
61	56	fveq2d 5885	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` Y ) )
62	49, 60, 61	3eltr4d 2532	. . . . . . 7 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) e. ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) )
63	47, 62	jaodan 792	. . . . . 6 |- ( ( ph /\ ( k = (/) \/ k = 1o ) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) e. ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) )
64	32, 63	sylan2 476	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) e. ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) )
65	18, 19, 20, 21, 22, 24, 27, 29, 64	prdsmet 21316	. . . 4 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) e. ( Met ` ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) ) )
66		xpscfn 15416	. . . . . . . 8 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
67	4, 5, 66	syl2anc 665	. . . . . . 7 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
68		dffn5 5926	. . . . . . 7 |- ( `' ( { R } +c { S } ) Fn 2o <-> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
69	67, 68	sylib 199	. . . . . 6 |- ( ph -> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
70	69	oveq2d 6321	. . . . 5 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
71	70	fveq2d 5885	. . . 4 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( dist ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
72	70	fveq2d 5885	. . . . . 6 |- ( ph -> ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
73	10, 72	eqtrd 2470	. . . . 5 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
74	73	fveq2d 5885	. . . 4 |- ( ph -> ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) = ( Met ` ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) ) )
75	65, 71, 74	3eltr4d 2532	. . 3 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
76		ssid 3489	. . 3 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
77		metres2 21309	. . 3 |- ( ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) /\ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) -> ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
78	75, 76, 77	sylancl 666	. 2 |- ( ph -> ( ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) |` ( ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) X. ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
79	9, 10, 13, 15, 16, 17, 78	imasf1omet 21322	1 |- ( ph -> P e. ( Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1870   _Vcvv 3087   C_ wss 3442   (/)c0 3767   {csn 4002   {cpr 4004   |-> cmpt 4484   X. cxp 4852   `'ccnv 4853   ran crn 4855   |` cres 4856   Fn wfn 5596   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   |-> cmpt2 6307   omcom 6706   1oc1o 7183   2oc2o 7184   Fincfn 7577   +c ccda 8595   Basecbs 15084  Scalarcsca 15155   distcds 15161   X__scprds 15303   X._s cxps 15363   Metcme 18891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-gsum 15300  df-prds 15305  df-xrs 15359  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-xmet 18898  df-met 18899

This theorem is referenced by: (None)