Theorem xpsmet 22187

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 2622	. . 3 |- ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
7		eqid 2622	. . 3 |- (Scalar ` R ) = (Scalar ` R )
8		eqid 2622	. . 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 16232	. 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 16233	. 2 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) )
11	6	xpsff1o2 16231	. . 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 6149	. . 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		ovexd 6680	. 2 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) e. _V )
15		eqid 2622	. 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 } ) ) ) )
16		xpsds.p	. 2 |- P = ( dist ` T )
17		eqid 2622	. . . . 5 |- ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
18		eqid 2622	. . . . 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 ) ) ) )
19		eqid 2622	. . . . 5 |- ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` ( `' ( { R } +c { S } ) ` k ) )
20		eqid 2622	. . . . 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 ) ) ) )
21		eqid 2622	. . . . 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 ) ) ) )
22		fvexd 6203	. . . . 5 |- ( ph -> (Scalar ` R ) e. _V )
23		2onn 7720	. . . . . 6 |- 2o e. om
24		nnfi 8153	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
25	23, 24	mp1i 13	. . . . 5 |- ( ph -> 2o e. Fin )
26		fvexd 6203	. . . . 5 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) e. _V )
27		elpri 4197	. . . . . . 7 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
28		df2o3 7573	. . . . . . 7 |- 2o = { (/) , 1o }
29	27, 28	eleq2s 2719	. . . . . 6 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
30		xpsmet.3	. . . . . . . . 9 |- ( ph -> M e. ( Met ` X ) )
31	30	adantr 481	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> M e. ( Met ` X ) )
32		fveq2 6191	. . . . . . . . . . . 12 |- ( k = (/) -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` (/) ) )
33		xpsc0 16220	. . . . . . . . . . . . 13 |- ( R e. V -> ( `' ( { R } +c { S } ) ` (/) ) = R )
34	4, 33	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` (/) ) = R )
35	32, 34	sylan9eqr 2678	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( `' ( { R } +c { S } ) ` k ) = R )
36	35	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` R ) )
37	35	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` R ) )
38	37, 2	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ph /\ k = (/) ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = X )
39	38	sqxpeqd 5141	. . . . . . . . . 10 |- ( ( ph /\ k = (/) ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( X X. X ) )
40	36, 39	reseq12d 5397	. . . . . . . . 9 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = ( ( dist ` R ) |` ( X X. X ) ) )
41		xpsds.m	. . . . . . . . 9 |- M = ( ( dist ` R ) |` ( X X. X ) )
42	40, 41	syl6eqr 2674	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = M )
43	38	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ k = (/) ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` X ) )
44	31, 42, 43	3eltr4d 2716	. . . . . . 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 ) ) ) )
45		xpsmet.4	. . . . . . . . 9 |- ( ph -> N e. ( Met ` Y ) )
46	45	adantr 481	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> N e. ( Met ` Y ) )
47		fveq2 6191	. . . . . . . . . . . 12 |- ( k = 1o -> ( `' ( { R } +c { S } ) ` k ) = ( `' ( { R } +c { S } ) ` 1o ) )
48		xpsc1 16221	. . . . . . . . . . . . 13 |- ( S e. W -> ( `' ( { R } +c { S } ) ` 1o ) = S )
49	5, 48	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' ( { R } +c { S } ) ` 1o ) = S )
50	47, 49	sylan9eqr 2678	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( `' ( { R } +c { S } ) ` k ) = S )
51	50	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( dist ` ( `' ( { R } +c { S } ) ` k ) ) = ( dist ` S ) )
52	50	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = ( Base ` S ) )
53	52, 3	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ph /\ k = 1o ) -> ( Base ` ( `' ( { R } +c { S } ) ` k ) ) = Y )
54	53	sqxpeqd 5141	. . . . . . . . . 10 |- ( ( ph /\ k = 1o ) -> ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Y X. Y ) )
55	51, 54	reseq12d 5397	. . . . . . . . 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 ) ) )
56		xpsds.n	. . . . . . . . 9 |- N = ( ( dist ` S ) |` ( Y X. Y ) )
57	55, 56	syl6eqr 2674	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( ( dist ` ( `' ( { R } +c { S } ) ` k ) ) |` ( ( Base ` ( `' ( { R } +c { S } ) ` k ) ) X. ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) ) = N )
58	53	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ k = 1o ) -> ( Met ` ( Base ` ( `' ( { R } +c { S } ) ` k ) ) ) = ( Met ` Y ) )
59	46, 57, 58	3eltr4d 2716	. . . . . . 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 ) ) ) )
60	44, 59	jaodan 826	. . . . . 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 ) ) ) )
61	29, 60	sylan2 491	. . . . 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 ) ) ) )
62	17, 18, 19, 20, 21, 22, 25, 26, 61	prdsmet 22175	. . . 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 ) ) ) ) ) )
63		xpscfn 16219	. . . . . . . 8 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
64	4, 5, 63	syl2anc 693	. . . . . . 7 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
65		dffn5 6241	. . . . . . 7 |- ( `' ( { R } +c { S } ) Fn 2o <-> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
66	64, 65	sylib 208	. . . . . 6 |- ( ph -> `' ( { R } +c { S } ) = ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) )
67	66	oveq2d 6666	. . . . 5 |- ( ph -> ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) = ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) )
68	67	fveq2d 6195	. . . 4 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( dist ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
69	67	fveq2d 6195	. . . . . 6 |- ( ph -> ( Base ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
70	10, 69	eqtrd 2656	. . . . 5 |- ( ph -> ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) = ( Base ` ( (Scalar ` R ) X_s ( k e. 2o |-> ( `' ( { R } +c { S } ) ` k ) ) ) ) )
71	70	fveq2d 6195	. . . 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 ) ) ) ) ) )
72	62, 68, 71	3eltr4d 2716	. . 3 |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
73		ssid 3624	. . 3 |- ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) C_ ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
74		metres2 22168	. . 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 } ) ) ) )
75	72, 73, 74	sylancl 694	. 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 } ) ) ) )
76	9, 10, 13, 14, 15, 16, 75	imasf1omet 22181	1 |- ( ph -> P e. ( Met ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1oc1o 7553   2oc2o 7554   Fincfn 7955   +c ccda 8989   Basecbs 15857  Scalarcsca 15944   distcds 15950   X__scprds 16106   X._s cxps 16166   Metcme 19732

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-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-0g 16102  df-gsum 16103  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-xmet 19739  df-met 19740

This theorem is referenced by: (None)