Theorem equivtotbnd 33577

Description: If the metric M is "strongly finer" than N (meaning that there is a positive real constant R such that N ( x , y ) <_ R x. M ( x , y )), then total boundedness of M implies total boundedness of N. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is totally bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
equivtotbnd.1	|- ( ph -> M e. ( TotBnd ` X ) )
equivtotbnd.2	|- ( ph -> N e. ( Met ` X ) )
equivtotbnd.3	|- ( ph -> R e. RR+ )
equivtotbnd.4	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x N y ) <_ ( R x. ( x M y ) ) )

Assertion

Ref	Expression
equivtotbnd	|- ( ph -> N e. ( TotBnd ` X ) )

Distinct variable groups:   x, y, M   x, N, y   ph, x, y   x, X, y   x, R, y

Proof of Theorem equivtotbnd

Dummy variables v s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		equivtotbnd.2	. 2 |- ( ph -> N e. ( Met ` X ) )
2		simpr 477	. . . . . 6 |- ( ( ph /\ r e. RR+ ) -> r e. RR+ )
3		equivtotbnd.3	. . . . . . 7 |- ( ph -> R e. RR+ )
4	3	adantr 481	. . . . . 6 |- ( ( ph /\ r e. RR+ ) -> R e. RR+ )
5	2, 4	rpdivcld 11889	. . . . 5 |- ( ( ph /\ r e. RR+ ) -> ( r / R ) e. RR+ )
6		equivtotbnd.1	. . . . . . 7 |- ( ph -> M e. ( TotBnd ` X ) )
7	6	adantr 481	. . . . . 6 |- ( ( ph /\ r e. RR+ ) -> M e. ( TotBnd ` X ) )
8		istotbnd3 33570	. . . . . . 7 |- ( M e. ( TotBnd ` X ) <-> ( M e. ( Met ` X ) /\ A. s e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) s ) = X ) )
9	8	simprbi 480	. . . . . 6 |- ( M e. ( TotBnd ` X ) -> A. s e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) s ) = X )
10	7, 9	syl 17	. . . . 5 |- ( ( ph /\ r e. RR+ ) -> A. s e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) s ) = X )
11		oveq2 6658	. . . . . . . . 9 |- ( s = ( r / R ) -> ( x ( ball ` M ) s ) = ( x ( ball ` M ) ( r / R ) ) )
12	11	iuneq2d 4547	. . . . . . . 8 |- ( s = ( r / R ) -> U_ x e. v ( x ( ball ` M ) s ) = U_ x e. v ( x ( ball ` M ) ( r / R ) ) )
13	12	eqeq1d 2624	. . . . . . 7 |- ( s = ( r / R ) -> ( U_ x e. v ( x ( ball ` M ) s ) = X <-> U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X ) )
14	13	rexbidv 3052	. . . . . 6 |- ( s = ( r / R ) -> ( E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) s ) = X <-> E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X ) )
15	14	rspcv 3305	. . . . 5 |- ( ( r / R ) e. RR+ -> ( A. s e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) s ) = X -> E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X ) )
16	5, 10, 15	sylc 65	. . . 4 |- ( ( ph /\ r e. RR+ ) -> E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X )
17		elfpw 8268	. . . . . . . . . . . . . 14 |- ( v e. ( ~P X i^i Fin ) <-> ( v C_ X /\ v e. Fin ) )
18	17	simplbi 476	. . . . . . . . . . . . 13 |- ( v e. ( ~P X i^i Fin ) -> v C_ X )
19	18	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> v C_ X )
20	19	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> x e. X )
21		eqid 2622	. . . . . . . . . . . . . 14 |- ( MetOpen ` N ) = ( MetOpen ` N )
22		eqid 2622	. . . . . . . . . . . . . 14 |- ( MetOpen ` M ) = ( MetOpen ` M )
23	8	simplbi 476	. . . . . . . . . . . . . . 15 |- ( M e. ( TotBnd ` X ) -> M e. ( Met ` X ) )
24	6, 23	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> M e. ( Met ` X ) )
25		equivtotbnd.4	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x N y ) <_ ( R x. ( x M y ) ) )
26	21, 22, 1, 24, 3, 25	metss2lem 22316	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) )
27	26	anass1rs 849	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. RR+ ) /\ x e. X ) -> ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) )
28	27	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. X ) -> ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) )
29	20, 28	syldan 487	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) )
30	29	ralrimiva 2966	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> A. x e. v ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) )
31		ss2iun 4536	. . . . . . . . 9 |- ( A. x e. v ( x ( ball ` M ) ( r / R ) ) C_ ( x ( ball ` N ) r ) -> U_ x e. v ( x ( ball ` M ) ( r / R ) ) C_ U_ x e. v ( x ( ball ` N ) r ) )
32	30, 31	syl 17	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> U_ x e. v ( x ( ball ` M ) ( r / R ) ) C_ U_ x e. v ( x ( ball ` N ) r ) )
33		sseq1 3626	. . . . . . . 8 |- ( U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X -> ( U_ x e. v ( x ( ball ` M ) ( r / R ) ) C_ U_ x e. v ( x ( ball ` N ) r ) <-> X C_ U_ x e. v ( x ( ball ` N ) r ) ) )
34	32, 33	syl5ibcom 235	. . . . . . 7 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> ( U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X -> X C_ U_ x e. v ( x ( ball ` N ) r ) ) )
35	1	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> N e. ( Met ` X ) )
36		metxmet 22139	. . . . . . . . . . 11 |- ( N e. ( Met ` X ) -> N e. ( *Met ` X ) )
37	35, 36	syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> N e. ( *Met ` X ) )
38		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> r e. RR+ )
39	38	rpxrd 11873	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> r e. RR* )
40		blssm 22223	. . . . . . . . . 10 |- ( ( N e. ( *Met ` X ) /\ x e. X /\ r e. RR* ) -> ( x ( ball ` N ) r ) C_ X )
41	37, 20, 39, 40	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) /\ x e. v ) -> ( x ( ball ` N ) r ) C_ X )
42	41	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> A. x e. v ( x ( ball ` N ) r ) C_ X )
43		iunss 4561	. . . . . . . 8 |- ( U_ x e. v ( x ( ball ` N ) r ) C_ X <-> A. x e. v ( x ( ball ` N ) r ) C_ X )
44	42, 43	sylibr 224	. . . . . . 7 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> U_ x e. v ( x ( ball ` N ) r ) C_ X )
45	34, 44	jctild 566	. . . . . 6 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> ( U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X -> ( U_ x e. v ( x ( ball ` N ) r ) C_ X /\ X C_ U_ x e. v ( x ( ball ` N ) r ) ) ) )
46		eqss 3618	. . . . . 6 |- ( U_ x e. v ( x ( ball ` N ) r ) = X <-> ( U_ x e. v ( x ( ball ` N ) r ) C_ X /\ X C_ U_ x e. v ( x ( ball ` N ) r ) ) )
47	45, 46	syl6ibr 242	. . . . 5 |- ( ( ( ph /\ r e. RR+ ) /\ v e. ( ~P X i^i Fin ) ) -> ( U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X -> U_ x e. v ( x ( ball ` N ) r ) = X ) )
48	47	reximdva 3017	. . . 4 |- ( ( ph /\ r e. RR+ ) -> ( E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` M ) ( r / R ) ) = X -> E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` N ) r ) = X ) )
49	16, 48	mpd 15	. . 3 |- ( ( ph /\ r e. RR+ ) -> E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` N ) r ) = X )
50	49	ralrimiva 2966	. 2 |- ( ph -> A. r e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` N ) r ) = X )
51		istotbnd3 33570	. 2 |- ( N e. ( TotBnd ` X ) <-> ( N e. ( Met ` X ) /\ A. r e. RR+ E. v e. ( ~P X i^i Fin ) U_ x e. v ( x ( ball ` N ) r ) = X ) )
52	1, 50, 51	sylanbrc 698	1 |- ( ph -> N e. ( TotBnd ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U_ciun 4520   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   x. cmul 9941   RR*cxr 10073   <_ cle 10075   / cdiv 10684   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736   TotBndctotbnd 33565

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-rp 11833  df-xadd 11947  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-totbnd 33567

This theorem is referenced by:  equivbnd2  33591