Theorem xpsmet 19957

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 2443	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2443	. . 3 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2443	. . 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 14510	. 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 14511	. 2 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 14509	. . 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 5653	. . 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 6116	. . 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 2443	. 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 2443	. . . . 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 2443	. . . . 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 2443	. . . . 5 |- ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` ( `' ( { R } +c { S } ) ` k ) )
21		eqid 2443	. . . . 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 2443	. . . . 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 5701	. . . . . 6 |- (Scalar ` R ) e. _V
24	23	a1i 11	. . . . 5 |- ( ph -> (Scalar ` R ) e. _V )
25		2onn 7079	. . . . . 6 |- 2o e. om
26		nnfi 7503	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
27	25, 26	mp1i 12	. . . . 5 |- ( ph -> 2o e. Fin )
28		fvex 5701	. . . . . 6 |- ( `' ( { R } +c { S } ) ` k ) e. _V
29	28	a1i 11	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) e. _V )
30		elpri 3897	. . . . . . 7 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
31		df2o3 6933	. . . . . . 7 |- 2o = { (/) , 1o }
32	30, 31	eleq2s 2535	. . . . . 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 5691	. . . . . . . . . . . 12 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
36		xpsc0 14498	. . . . . . . . . . . . 13 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
37	4, 36	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
38	35, 37	sylan9eqr 2497	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( `' ( { R } +c { S } ) ` k ) = R )
39	38	fveq2d 5695	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` R ) )
40	38	fveq2d 5695	. . . . . . . . . . . 12 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` R ) )
41	40, 2	syl6eqr 2493	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = X )
42	41, 41	xpeq12d 4865	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( X X. X ) )
43	39, 42	reseq12d 5111	. . . . . . . . 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 2493	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = M )
46	41	fveq2d 5695	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` X ) )
47	34, 45, 46	3eltr4d 2524	. . . . . . 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 5691	. . . . . . . . . . . 12 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
51		xpsc1 14499	. . . . . . . . . . . . 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 2497	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( `' ( { R } +c { S } ) ` k ) = S )
54	53	fveq2d 5695	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` S ) )
55	53	fveq2d 5695	. . . . . . . . . . . 12 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` S ) )
56	55, 3	syl6eqr 2493	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = Y )
57	56, 56	xpeq12d 4865	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Y X. Y ) )
58	54, 57	reseq12d 5111	. . . . . . . . 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 2493	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = N )
61	56	fveq2d 5695	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` Y ) )
62	49, 60, 61	3eltr4d 2524	. . . . . . 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 19945	. . . 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 14497	. . . . . . . 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 5737	. . . . . . 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 6107	. . . . 5 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
71	70	fveq2d 5695	. . . 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 5695	. . . . . 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 2475	. . . . 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 5695	. . . 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 2524	. . 3 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
76		ssid 3375	. . 3 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
77		metres2 19938	. . 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 19951	1 |- ( ph -> P e. ( Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2972   C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879   e. cmpt 4350   X. cxp 4838   `'ccnv 4839   ran crn 4841   |` cres 4842   Fn wfn 5413   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   e. cmpt2 6093   omcom 6476   1oc1o 6913   2oc2o 6914   Fincfn 7310   +c ccda 8336   Basecbs 14174  Scalarcsca 14241   distcds 14247   X__scprds 14384   X._s cxps 14444   Metcme 17802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-xrs 14440  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-xmet 17810  df-met 17811

This theorem is referenced by: (None)