Theorem xpsmet 20648

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 2467	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2467	. . 3 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2467	. . 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 14827	. 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 14828	. 2 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 14826	. . 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 5828	. . 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 12	. 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 6309	. . 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 2467	. 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 2467	. . . . 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 2467	. . . . 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 2467	. . . . 5 |- ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` ( `' ( { R } +c { S } ) ` k ) )
21		eqid 2467	. . . . 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 2467	. . . . 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 5876	. . . . . 6 |- (Scalar ` R ) e. _V
24	23	a1i 11	. . . . 5 |- ( ph -> (Scalar ` R ) e. _V )
25		2onn 7289	. . . . . 6 |- 2o e. om
26		nnfi 7710	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
27	25, 26	mp1i 12	. . . . 5 |- ( ph -> 2o e. Fin )
28		fvex 5876	. . . . . 6 |- ( `' ( { R } +c { S } ) ` k ) e. _V
29	28	a1i 11	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) e. _V )
30		elpri 4047	. . . . . . 7 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
31		df2o3 7143	. . . . . . 7 |- 2o = { (/) , 1o }
32	30, 31	eleq2s 2575	. . . . . 6 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
33		xpsmet.3	. . . . . . . . 9 |- ( ph -> M e. ( Met ` X ) )
34	33	adantr 465	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> M e. ( Met ` X ) )
35		fveq2 5866	. . . . . . . . . . . 12 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
36		xpsc0 14815	. . . . . . . . . . . . 13 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
37	4, 36	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
38	35, 37	sylan9eqr 2530	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( `' ( { R } +c { S } ) ` k ) = R )
39	38	fveq2d 5870	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` R ) )
40	38	fveq2d 5870	. . . . . . . . . . . 12 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` R ) )
41	40, 2	syl6eqr 2526	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = X )
42	41, 41	xpeq12d 5024	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( X X. X ) )
43	39, 42	reseq12d 5274	. . . . . . . . 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 2526	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = M )
46	41	fveq2d 5870	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` X ) )
47	34, 45, 46	3eltr4d 2570	. . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> N e. ( Met ` Y ) )
50		fveq2 5866	. . . . . . . . . . . 12 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
51		xpsc1 14816	. . . . . . . . . . . . 13 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
52	5, 51	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
53	50, 52	sylan9eqr 2530	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( `' ( { R } +c { S } ) ` k ) = S )
54	53	fveq2d 5870	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` S ) )
55	53	fveq2d 5870	. . . . . . . . . . . 12 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` S ) )
56	55, 3	syl6eqr 2526	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = Y )
57	56, 56	xpeq12d 5024	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Y X. Y ) )
58	54, 57	reseq12d 5274	. . . . . . . . 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 2526	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = N )
61	56	fveq2d 5870	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` Y ) )
62	49, 60, 61	3eltr4d 2570	. . . . . . 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 783	. . . . . 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 474	. . . . 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 20636	. . . 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 14814	. . . . . . . 8 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
67	4, 5, 66	syl2anc 661	. . . . . . 7 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
68		dffn5 5913	. . . . . . 7 |- ( `' ( { R } +c { S } ) Fn 2o <-> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
69	67, 68	sylib 196	. . . . . 6 |- ( ph -> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
70	69	oveq2d 6300	. . . . 5 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
71	70	fveq2d 5870	. . . 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 5870	. . . . . 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 2508	. . . . 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 5870	. . . 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 2570	. . 3 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
76		ssid 3523	. . 3 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
77		metres2 20629	. . 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 662	. 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 20642	1 |- ( ph -> P e. ( Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   |-> cmpt 4505   X. cxp 4997   `'ccnv 4998   ran crn 5000   |` cres 5001   Fn wfn 5583   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   |-> cmpt2 6286   omcom 6684   1oc1o 7123   2oc2o 7124   Fincfn 7516   +c ccda 8547   Basecbs 14490  Scalarcsca 14558   distcds 14564   X__scprds 14701   X._s cxps 14761   Metcme 18203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-icc 11536  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-hom 14579  df-cco 14580  df-0g 14697  df-gsum 14698  df-prds 14703  df-xrs 14757  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-xmet 18211  df-met 18212

This theorem is referenced by: (None)