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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 8002 |
. . 3
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3 | expcnvre.a1 |
. . 3
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4 | qbtwnre 10287 |
. . 3
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5 | 1, 2, 3, 4 | syl3anc 1249 |
. 2
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6 | nn0uz 9592 |
. . 3
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7 | 0zd 9295 |
. . 3
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8 | qre 9655 |
. . . . . 6
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9 | 8 | ad2antrl 490 |
. . . . 5
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10 | 9 | recnd 8016 |
. . . 4
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11 | 0red 7988 |
. . . . . . 7
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12 | 1 | adantr 276 |
. . . . . . . 8
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13 | expcnvre.a0 |
. . . . . . . . 9
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14 | 13 | adantr 276 |
. . . . . . . 8
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15 | simprrl 539 |
. . . . . . . 8
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16 | 11, 12, 9, 14, 15 | lelttrd 8112 |
. . . . . . 7
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17 | 11, 9, 16 | ltled 8106 |
. . . . . 6
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18 | 9, 17 | absidd 11208 |
. . . . 5
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19 | simprrr 540 |
. . . . 5
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20 | 18, 19 | eqbrtrd 4040 |
. . . 4
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21 | 9, 16 | gt0ap0d 8616 |
. . . 4
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22 | 10, 20, 21 | expcnvap0 11542 |
. . 3
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23 | nn0ex 9212 |
. . . . 5
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24 | 23 | mptex 5763 |
. . . 4
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25 | 24 | a1i 9 |
. . 3
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26 | simpr 110 |
. . . . 5
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27 | 9 | adantr 276 |
. . . . . 6
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28 | 27, 26 | reexpcld 10702 |
. . . . 5
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29 | oveq2 5904 |
. . . . . 6
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30 | eqid 2189 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | fvmptg 5613 |
. . . . 5
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32 | 26, 28, 31 | syl2anc 411 |
. . . 4
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33 | 32, 28 | eqeltrd 2266 |
. . 3
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34 | 12 | adantr 276 |
. . . . . 6
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35 | 34, 26 | reexpcld 10702 |
. . . . 5
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36 | oveq2 5904 |
. . . . . 6
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37 | eqid 2189 |
. . . . . 6
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38 | 36, 37 | fvmptg 5613 |
. . . . 5
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39 | 26, 35, 38 | syl2anc 411 |
. . . 4
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40 | 39, 35 | eqeltrd 2266 |
. . 3
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41 | 14 | adantr 276 |
. . . . 5
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42 | 15 | adantr 276 |
. . . . . 6
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43 | 34, 27, 42 | ltled 8106 |
. . . . 5
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44 | leexp1a 10606 |
. . . . 5
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45 | 34, 27, 26, 41, 43, 44 | syl32anc 1257 |
. . . 4
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46 | 45, 39, 32 | 3brtr4d 4050 |
. . 3
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47 | 34, 26, 41 | expge0d 10703 |
. . . 4
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48 | 47, 39 | breqtrrd 4046 |
. . 3
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49 | 6, 7, 22, 25, 33, 40, 46, 48 | climsqz2 11376 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 5, 49 | rexlimddv 2612 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-seqfrec 10477 df-exp 10551 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-clim 11319 |
This theorem is referenced by: expcnv 11544 |
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