Theorem xpsmet 21475

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 2471	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2471	. . 3 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2471	. . 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 15556	. 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 15557	. 2 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 15555	. . 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 5840	. . 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 6336	. . 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 2471	. 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 2471	. . . . 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 2471	. . . . 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 2471	. . . . 5 |- ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` ( `' ( { R } +c { S } ) ` k ) )
21		eqid 2471	. . . . 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 2471	. . . . 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 5889	. . . . . 6 |- (Scalar ` R ) e. _V
24	23	a1i 11	. . . . 5 |- ( ph -> (Scalar ` R ) e. _V )
25		2onn 7359	. . . . . 6 |- 2o e. om
26		nnfi 7783	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
27	25, 26	mp1i 13	. . . . 5 |- ( ph -> 2o e. Fin )
28		fvex 5889	. . . . . 6 |- ( `' ( { R } +c { S } ) ` k ) e. _V
29	28	a1i 11	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) e. _V )
30		elpri 3976	. . . . . . 7 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
31		df2o3 7213	. . . . . . 7 |- 2o = { (/) , 1o }
32	30, 31	eleq2s 2567	. . . . . 6 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
33		xpsmet.3	. . . . . . . . 9 |- ( ph -> M e. ( Met ` X ) )
34	33	adantr 472	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> M e. ( Met ` X ) )
35		fveq2 5879	. . . . . . . . . . . 12 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
36		xpsc0 15544	. . . . . . . . . . . . 13 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
37	4, 36	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
38	35, 37	sylan9eqr 2527	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( `' ( { R } +c { S } ) ` k ) = R )
39	38	fveq2d 5883	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` R ) )
40	38	fveq2d 5883	. . . . . . . . . . . 12 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` R ) )
41	40, 2	syl6eqr 2523	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = X )
42	41	sqxpeqd 4865	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( X X. X ) )
43	39, 42	reseq12d 5112	. . . . . . . . 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 2523	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = M )
46	41	fveq2d 5883	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` X ) )
47	34, 45, 46	3eltr4d 2564	. . . . . . 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 472	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> N e. ( Met ` Y ) )
50		fveq2 5879	. . . . . . . . . . . 12 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
51		xpsc1 15545	. . . . . . . . . . . . 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 2527	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( `' ( { R } +c { S } ) ` k ) = S )
54	53	fveq2d 5883	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` S ) )
55	53	fveq2d 5883	. . . . . . . . . . . 12 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` S ) )
56	55, 3	syl6eqr 2523	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = Y )
57	56	sqxpeqd 4865	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Y X. Y ) )
58	54, 57	reseq12d 5112	. . . . . . . . 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 2523	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = N )
61	56	fveq2d 5883	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` Y ) )
62	49, 60, 61	3eltr4d 2564	. . . . . . 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 802	. . . . . 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 482	. . . . 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 21463	. . . 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 15543	. . . . . . . 8 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
67	4, 5, 66	syl2anc 673	. . . . . . 7 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
68		dffn5 5924	. . . . . . 7 |- ( `' ( { R } +c { S } ) Fn 2o <-> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
69	67, 68	sylib 201	. . . . . 6 |- ( ph -> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
70	69	oveq2d 6324	. . . . 5 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
71	70	fveq2d 5883	. . . 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 5883	. . . . . 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 2505	. . . . 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 5883	. . . 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 2564	. . 3 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
76		ssid 3437	. . 3 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
77		metres2 21456	. . 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 675	. 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 21469	1 |- ( ph -> P e. ( Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   ran crn 4840   |` cres 4841   Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   omcom 6711   1oc1o 7193   2oc2o 7194   Fincfn 7587   +c ccda 8615   Basecbs 15199  Scalarcsca 15271   distcds 15277   X__scprds 15422   X._s cxps 15483   Metcme 19033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-xrs 15478  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-xmet 19040  df-met 19041

This theorem is referenced by: (None)