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Mirrors > Home > ILE Home > Th. List > cnrehmeocntop | Unicode version |
Description: The canonical bijection from to described in cnref1o 9440 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
cnrehmeo.1 | |
cnrehmeo.2 | |
cnrehmeocntop.3 |
Ref | Expression |
---|---|
cnrehmeocntop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnrehmeo.1 | . . . 4 | |
2 | cnrehmeo.2 | . . . . . . 7 | |
3 | retopon 12695 | . . . . . . 7 TopOn | |
4 | 2, 3 | eqeltri 2212 | . . . . . 6 TopOn |
5 | 4 | a1i 9 | . . . . 5 TopOn |
6 | cnrehmeocntop.3 | . . . . . . . 8 | |
7 | 6 | cntoptop 12702 | . . . . . . 7 |
8 | cnrest2r 12406 | . . . . . . 7 ↾t | |
9 | 7, 8 | mp1i 10 | . . . . . 6 ↾t |
10 | 5, 5 | cnmpt1st 12457 | . . . . . . 7 |
11 | 6 | tgioo2cntop 12718 | . . . . . . . . 9 ↾t |
12 | 2, 11 | eqtri 2160 | . . . . . . . 8 ↾t |
13 | 12 | oveq2i 5785 | . . . . . . 7 ↾t |
14 | 10, 13 | eleqtrdi 2232 | . . . . . 6 ↾t |
15 | 9, 14 | sseldd 3098 | . . . . 5 |
16 | 6 | cntoptopon 12701 | . . . . . . . 8 TopOn |
17 | 16 | a1i 9 | . . . . . . 7 TopOn |
18 | ax-icn 7715 | . . . . . . . 8 | |
19 | 18 | a1i 9 | . . . . . . 7 |
20 | 5, 5, 17, 19 | cnmpt2c 12459 | . . . . . 6 |
21 | 5, 5 | cnmpt2nd 12458 | . . . . . . . 8 |
22 | 21, 13 | eleqtrdi 2232 | . . . . . . 7 ↾t |
23 | 9, 22 | sseldd 3098 | . . . . . 6 |
24 | 6 | mulcncntop 12723 | . . . . . . 7 |
25 | 24 | a1i 9 | . . . . . 6 |
26 | 5, 5, 20, 23, 25 | cnmpt22f 12464 | . . . . 5 |
27 | 6 | addcncntop 12721 | . . . . . 6 |
28 | 27 | a1i 9 | . . . . 5 |
29 | 5, 5, 15, 26, 28 | cnmpt22f 12464 | . . . 4 |
30 | 1, 29 | eqeltrid 2226 | . . 3 |
31 | 1 | cnrecnv 10682 | . . . 4 |
32 | ref 10627 | . . . . . . . 8 | |
33 | 32 | a1i 9 | . . . . . . 7 |
34 | 33 | feqmptd 5474 | . . . . . 6 |
35 | recncf 12742 | . . . . . . 7 | |
36 | ssid 3117 | . . . . . . . 8 | |
37 | ax-resscn 7712 | . . . . . . . 8 | |
38 | 16 | toponrestid 12188 | . . . . . . . . 9 ↾t |
39 | 6, 38, 12 | cncfcncntop 12749 | . . . . . . . 8 |
40 | 36, 37, 39 | mp2an 422 | . . . . . . 7 |
41 | 35, 40 | eleqtri 2214 | . . . . . 6 |
42 | 34, 41 | eqeltrrdi 2231 | . . . . 5 |
43 | imf 10628 | . . . . . . . 8 | |
44 | 43 | a1i 9 | . . . . . . 7 |
45 | 44 | feqmptd 5474 | . . . . . 6 |
46 | imcncf 12743 | . . . . . . 7 | |
47 | 46, 40 | eleqtri 2214 | . . . . . 6 |
48 | 45, 47 | eqeltrrdi 2231 | . . . . 5 |
49 | 17, 42, 48 | cnmpt1t 12454 | . . . 4 |
50 | 31, 49 | eqeltrid 2226 | . . 3 |
51 | ishmeo 12473 | . . 3 | |
52 | 30, 50, 51 | sylanbrc 413 | . 2 |
53 | 52 | mptru 1340 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wtru 1332 wcel 1480 wss 3071 cop 3530 cmpt 3989 ccnv 4538 crn 4540 ccom 4543 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 cc 7618 cr 7619 ci 7622 caddc 7623 cmul 7625 cmin 7933 cioo 9671 cre 10612 cim 10613 cabs 10769 ↾t crest 12120 ctg 12135 cmopn 12154 ctop 12164 TopOnctopon 12177 ccn 12354 ctx 12421 chmeo 12469 ccncf 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 ax-addf 7742 ax-mulf 7743 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-ioo 9675 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-rest 12122 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-met 12158 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-cnp 12358 df-tx 12422 df-hmeo 12470 df-cncf 12727 |
This theorem is referenced by: (None) |
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