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Mirrors > Home > MPE Home > Th. List > Mathboxes > equivbnd | Structured version Visualization version Unicode version |
Description: If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), then boundedness of implies boundedness of . (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.) |
Ref | Expression |
---|---|
equivbnd.1 | |
equivbnd.2 | |
equivbnd.3 | |
equivbnd.4 |
Ref | Expression |
---|---|
equivbnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equivbnd.2 | . 2 | |
2 | equivbnd.1 | . . . 4 | |
3 | isbnd3b 33584 | . . . . 5 | |
4 | 3 | simprbi 480 | . . . 4 |
5 | 2, 4 | syl 17 | . . 3 |
6 | equivbnd.3 | . . . . . . 7 | |
7 | 6 | rpred 11872 | . . . . . 6 |
8 | remulcl 10021 | . . . . . 6 | |
9 | 7, 8 | sylan 488 | . . . . 5 |
10 | bndmet 33580 | . . . . . . . . . . 11 | |
11 | 2, 10 | syl 17 | . . . . . . . . . 10 |
12 | 11 | adantr 481 | . . . . . . . . 9 |
13 | metcl 22137 | . . . . . . . . . 10 | |
14 | 13 | 3expb 1266 | . . . . . . . . 9 |
15 | 12, 14 | sylan 488 | . . . . . . . 8 |
16 | simplr 792 | . . . . . . . 8 | |
17 | 6 | ad2antrr 762 | . . . . . . . 8 |
18 | 15, 16, 17 | lemul2d 11916 | . . . . . . 7 |
19 | equivbnd.4 | . . . . . . . . 9 | |
20 | 19 | adantlr 751 | . . . . . . . 8 |
21 | 1 | adantr 481 | . . . . . . . . . 10 |
22 | metcl 22137 | . . . . . . . . . . 11 | |
23 | 22 | 3expb 1266 | . . . . . . . . . 10 |
24 | 21, 23 | sylan 488 | . . . . . . . . 9 |
25 | 7 | ad2antrr 762 | . . . . . . . . . 10 |
26 | 25, 15 | remulcld 10070 | . . . . . . . . 9 |
27 | 9 | adantr 481 | . . . . . . . . 9 |
28 | letr 10131 | . . . . . . . . 9 | |
29 | 24, 26, 27, 28 | syl3anc 1326 | . . . . . . . 8 |
30 | 20, 29 | mpand 711 | . . . . . . 7 |
31 | 18, 30 | sylbid 230 | . . . . . 6 |
32 | 31 | ralimdvva 2964 | . . . . 5 |
33 | breq2 4657 | . . . . . . 7 | |
34 | 33 | 2ralbidv 2989 | . . . . . 6 |
35 | 34 | rspcev 3309 | . . . . 5 |
36 | 9, 32, 35 | syl6an 568 | . . . 4 |
37 | 36 | rexlimdva 3031 | . . 3 |
38 | 5, 37 | mpd 15 | . 2 |
39 | isbnd3b 33584 | . 2 | |
40 | 1, 38, 39 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 class class class wbr 4653 cfv 5888 (class class class)co 6650 cr 9935 cmul 9941 cle 10075 crp 11832 cme 19732 cbnd 33566 |
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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-1st 7168 df-2nd 7169 df-er 7742 df-ec 7744 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 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-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-bnd 33578 |
This theorem is referenced by: equivbnd2 33591 |
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