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Mirrors > Home > ILE Home > Th. List > expnbnd | Unicode version |
Description: Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.) |
Ref | Expression |
---|---|
expnbnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | simp2 982 | . . . 4 | |
4 | 3 | adantr 274 | . . 3 |
5 | simpr 109 | . . 3 | |
6 | simp3 983 | . . . 4 | |
7 | 6 | adantr 274 | . . 3 |
8 | 1red 7774 | . . . . . . . . 9 | |
9 | 1, 8 | resubcld 8136 | . . . . . . . 8 |
10 | 3, 8 | resubcld 8136 | . . . . . . . 8 |
11 | 8, 3 | posdifd 8287 | . . . . . . . . . 10 |
12 | 6, 11 | mpbid 146 | . . . . . . . . 9 |
13 | 10, 12 | gt0ap0d 8384 | . . . . . . . 8 # |
14 | 9, 10, 13 | redivclapd 8587 | . . . . . . 7 |
15 | arch 8967 | . . . . . . 7 | |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | 16 | 3expa 1181 | . . . . 5 |
18 | 17 | adantrl 469 | . . . 4 |
19 | simplll 522 | . . . . . . . 8 | |
20 | 19 | adantr 274 | . . . . . . 7 |
21 | simpllr 523 | . . . . . . . . . . 11 | |
22 | 1red 7774 | . . . . . . . . . . 11 | |
23 | 21, 22 | resubcld 8136 | . . . . . . . . . 10 |
24 | simpr 109 | . . . . . . . . . . 11 | |
25 | 24 | nnred 8726 | . . . . . . . . . 10 |
26 | 23, 25 | remulcld 7789 | . . . . . . . . 9 |
27 | 26, 22 | readdcld 7788 | . . . . . . . 8 |
28 | 27 | adantr 274 | . . . . . . 7 |
29 | 24 | nnnn0d 9023 | . . . . . . . . 9 |
30 | reexpcl 10303 | . . . . . . . . 9 | |
31 | 21, 29, 30 | syl2anc 408 | . . . . . . . 8 |
32 | 31 | adantr 274 | . . . . . . 7 |
33 | simpr 109 | . . . . . . . . 9 | |
34 | 1red 7774 | . . . . . . . . . . 11 | |
35 | 20, 34 | resubcld 8136 | . . . . . . . . . 10 |
36 | simplr 519 | . . . . . . . . . . 11 | |
37 | 36 | nnred 8726 | . . . . . . . . . 10 |
38 | 21 | adantr 274 | . . . . . . . . . . 11 |
39 | 38, 34 | resubcld 8136 | . . . . . . . . . 10 |
40 | simplrr 525 | . . . . . . . . . . . 12 | |
41 | 40 | adantr 274 | . . . . . . . . . . 11 |
42 | 34, 38 | posdifd 8287 | . . . . . . . . . . 11 |
43 | 41, 42 | mpbid 146 | . . . . . . . . . 10 |
44 | ltdivmul 8627 | . . . . . . . . . 10 | |
45 | 35, 37, 39, 43, 44 | syl112anc 1220 | . . . . . . . . 9 |
46 | 33, 45 | mpbid 146 | . . . . . . . 8 |
47 | 39, 37 | remulcld 7789 | . . . . . . . . 9 |
48 | 20, 34, 47 | ltsubaddd 8296 | . . . . . . . 8 |
49 | 46, 48 | mpbid 146 | . . . . . . 7 |
50 | 36 | nnnn0d 9023 | . . . . . . . 8 |
51 | 0red 7760 | . . . . . . . . . 10 | |
52 | 0lt1 7882 | . . . . . . . . . . . 12 | |
53 | 0re 7759 | . . . . . . . . . . . . 13 | |
54 | 1re 7758 | . . . . . . . . . . . . 13 | |
55 | lttr 7831 | . . . . . . . . . . . . 13 | |
56 | 53, 54, 55 | mp3an12 1305 | . . . . . . . . . . . 12 |
57 | 52, 56 | mpani 426 | . . . . . . . . . . 11 |
58 | 21, 40, 57 | sylc 62 | . . . . . . . . . 10 |
59 | 51, 21, 58 | ltled 7874 | . . . . . . . . 9 |
60 | 59 | adantr 274 | . . . . . . . 8 |
61 | bernneq2 10406 | . . . . . . . 8 | |
62 | 38, 50, 60, 61 | syl3anc 1216 | . . . . . . 7 |
63 | 20, 28, 32, 49, 62 | ltletrd 8178 | . . . . . 6 |
64 | 63 | ex 114 | . . . . 5 |
65 | 64 | reximdva 2532 | . . . 4 |
66 | 18, 65 | mpd 13 | . . 3 |
67 | 2, 4, 5, 7, 66 | syl22anc 1217 | . 2 |
68 | 1nn 8724 | . . 3 | |
69 | simpr 109 | . . . 4 | |
70 | simpl2 985 | . . . . . 6 | |
71 | 70 | recnd 7787 | . . . . 5 |
72 | exp1 10292 | . . . . 5 | |
73 | 71, 72 | syl 14 | . . . 4 |
74 | 69, 73 | breqtrrd 3951 | . . 3 |
75 | oveq2 5775 | . . . . 5 | |
76 | 75 | breq2d 3936 | . . . 4 |
77 | 76 | rspcev 2784 | . . 3 |
78 | 68, 74, 77 | sylancr 410 | . 2 |
79 | axltwlin 7825 | . . . . 5 | |
80 | 54, 79 | mp3an1 1302 | . . . 4 |
81 | 80 | ancoms 266 | . . 3 |
82 | 81 | 3impia 1178 | . 2 |
83 | 67, 78, 82 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 caddc 7616 cmul 7618 clt 7793 cle 7794 cmin 7926 cdiv 8425 cn 8713 cn0 8970 cexp 10285 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-exp 10286 |
This theorem is referenced by: expnlbnd 10409 |
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