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Mirrors > Home > ILE Home > Th. List > expcnvre | Unicode version |
Description: A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.) |
Ref | Expression |
---|---|
expcnvre.ar |
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expcnvre.a1 |
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expcnvre.a0 |
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Ref | Expression |
---|---|
expcnvre |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcnvre.ar |
. . 3
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2 | 1red 7557 |
. . 3
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3 | expcnvre.a1 |
. . 3
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4 | qbtwnre 9722 |
. . 3
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5 | 1, 2, 3, 4 | syl3anc 1175 |
. 2
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6 | nn0uz 9107 |
. . 3
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7 | 0zd 8816 |
. . 3
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8 | qre 9164 |
. . . . . 6
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9 | 8 | ad2antrl 475 |
. . . . 5
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10 | 9 | recnd 7570 |
. . . 4
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11 | 0red 7543 |
. . . . . . 7
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12 | 1 | adantr 271 |
. . . . . . . 8
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13 | expcnvre.a0 |
. . . . . . . . 9
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14 | 13 | adantr 271 |
. . . . . . . 8
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15 | simprrl 507 |
. . . . . . . 8
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16 | 11, 12, 9, 14, 15 | lelttrd 7662 |
. . . . . . 7
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17 | 11, 9, 16 | ltled 7656 |
. . . . . 6
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18 | 9, 17 | absidd 10654 |
. . . . 5
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19 | simprrr 508 |
. . . . 5
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20 | 18, 19 | eqbrtrd 3871 |
. . . 4
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21 | 9, 16 | gt0ap0d 8159 |
. . . 4
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22 | 10, 20, 21 | expcnvap0 10950 |
. . 3
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23 | nn0ex 8733 |
. . . . 5
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24 | 23 | mptex 5537 |
. . . 4
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25 | 24 | a1i 9 |
. . 3
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26 | simpr 109 |
. . . . 5
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27 | 9 | adantr 271 |
. . . . . 6
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28 | 27, 26 | reexpcld 10157 |
. . . . 5
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29 | oveq2 5674 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | eqid 2089 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | fvmptg 5393 |
. . . . 5
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32 | 26, 28, 31 | syl2anc 404 |
. . . 4
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33 | 32, 28 | eqeltrd 2165 |
. . 3
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34 | 12 | adantr 271 |
. . . . . 6
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35 | 34, 26 | reexpcld 10157 |
. . . . 5
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36 | oveq2 5674 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | eqid 2089 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 36, 37 | fvmptg 5393 |
. . . . 5
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39 | 26, 35, 38 | syl2anc 404 |
. . . 4
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40 | 39, 35 | eqeltrd 2165 |
. . 3
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41 | 14 | adantr 271 |
. . . . 5
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42 | 15 | adantr 271 |
. . . . . 6
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43 | 34, 27, 42 | ltled 7656 |
. . . . 5
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44 | leexp1a 10064 |
. . . . 5
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45 | 34, 27, 26, 41, 43, 44 | syl32anc 1183 |
. . . 4
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46 | 45, 39, 32 | 3brtr4d 3881 |
. . 3
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47 | 34, 26, 41 | expge0d 10158 |
. . . 4
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48 | 47, 39 | breqtrrd 3877 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 6, 7, 22, 25, 33, 40, 46, 48 | climsqz2 10778 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 5, 49 | rexlimddv 2494 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-iseq 9907 df-seq3 9908 df-exp 10009 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 |
This theorem is referenced by: expcnv 10952 |
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