Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 938 | . . . 4 | |
2 | 1 | adantr 270 | . . 3 |
3 | simp2 939 | . . . 4 | |
4 | 3 | adantr 270 | . . 3 |
5 | simpr 108 | . . 3 | |
6 | simp3 940 | . . . 4 | |
7 | 6 | adantr 270 | . . 3 |
8 | 1red 7134 | . . . . . . . . 9 | |
9 | 1, 8 | resubcld 7485 | . . . . . . . 8 |
10 | 3, 8 | resubcld 7485 | . . . . . . . 8 |
11 | 8, 3 | posdifd 7632 | . . . . . . . . . 10 |
12 | 6, 11 | mpbid 145 | . . . . . . . . 9 |
13 | 10, 12 | gt0ap0d 7728 | . . . . . . . 8 # |
14 | 9, 10, 13 | redivclapd 7920 | . . . . . . 7 |
15 | arch 8285 | . . . . . . 7 | |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | 16 | 3expa 1138 | . . . . 5 |
18 | 17 | adantrl 461 | . . . 4 |
19 | simplll 499 | . . . . . . . 8 | |
20 | 19 | adantr 270 | . . . . . . 7 |
21 | simpllr 500 | . . . . . . . . . . 11 | |
22 | 1red 7134 | . . . . . . . . . . 11 | |
23 | 21, 22 | resubcld 7485 | . . . . . . . . . 10 |
24 | simpr 108 | . . . . . . . . . . 11 | |
25 | 24 | nnred 8052 | . . . . . . . . . 10 |
26 | 23, 25 | remulcld 7149 | . . . . . . . . 9 |
27 | 26, 22 | readdcld 7148 | . . . . . . . 8 |
28 | 27 | adantr 270 | . . . . . . 7 |
29 | 24 | nnnn0d 8341 | . . . . . . . . 9 |
30 | reexpcl 9493 | . . . . . . . . 9 | |
31 | 21, 29, 30 | syl2anc 403 | . . . . . . . 8 |
32 | 31 | adantr 270 | . . . . . . 7 |
33 | simpr 108 | . . . . . . . . 9 | |
34 | 1red 7134 | . . . . . . . . . . 11 | |
35 | 20, 34 | resubcld 7485 | . . . . . . . . . 10 |
36 | simplr 496 | . . . . . . . . . . 11 | |
37 | 36 | nnred 8052 | . . . . . . . . . 10 |
38 | 21 | adantr 270 | . . . . . . . . . . 11 |
39 | 38, 34 | resubcld 7485 | . . . . . . . . . 10 |
40 | simplrr 502 | . . . . . . . . . . . 12 | |
41 | 40 | adantr 270 | . . . . . . . . . . 11 |
42 | 34, 38 | posdifd 7632 | . . . . . . . . . . 11 |
43 | 41, 42 | mpbid 145 | . . . . . . . . . 10 |
44 | ltdivmul 7954 | . . . . . . . . . 10 | |
45 | 35, 37, 39, 43, 44 | syl112anc 1173 | . . . . . . . . 9 |
46 | 33, 45 | mpbid 145 | . . . . . . . 8 |
47 | 39, 37 | remulcld 7149 | . . . . . . . . 9 |
48 | 20, 34, 47 | ltsubaddd 7641 | . . . . . . . 8 |
49 | 46, 48 | mpbid 145 | . . . . . . 7 |
50 | 36 | nnnn0d 8341 | . . . . . . . 8 |
51 | 0red 7120 | . . . . . . . . . 10 | |
52 | 0lt1 7236 | . . . . . . . . . . . 12 | |
53 | 0re 7119 | . . . . . . . . . . . . 13 | |
54 | 1re 7118 | . . . . . . . . . . . . 13 | |
55 | lttr 7185 | . . . . . . . . . . . . 13 | |
56 | 53, 54, 55 | mp3an12 1258 | . . . . . . . . . . . 12 |
57 | 52, 56 | mpani 420 | . . . . . . . . . . 11 |
58 | 21, 40, 57 | sylc 61 | . . . . . . . . . 10 |
59 | 51, 21, 58 | ltled 7228 | . . . . . . . . 9 |
60 | 59 | adantr 270 | . . . . . . . 8 |
61 | bernneq2 9594 | . . . . . . . 8 | |
62 | 38, 50, 60, 61 | syl3anc 1169 | . . . . . . 7 |
63 | 20, 28, 32, 49, 62 | ltletrd 7527 | . . . . . 6 |
64 | 63 | ex 113 | . . . . 5 |
65 | 64 | reximdva 2463 | . . . 4 |
66 | 18, 65 | mpd 13 | . . 3 |
67 | 2, 4, 5, 7, 66 | syl22anc 1170 | . 2 |
68 | 1nn 8050 | . . 3 | |
69 | simpr 108 | . . . 4 | |
70 | simpl2 942 | . . . . . 6 | |
71 | 70 | recnd 7147 | . . . . 5 |
72 | exp1 9482 | . . . . 5 | |
73 | 71, 72 | syl 14 | . . . 4 |
74 | 69, 73 | breqtrrd 3811 | . . 3 |
75 | oveq2 5540 | . . . . 5 | |
76 | 75 | breq2d 3797 | . . . 4 |
77 | 76 | rspcev 2701 | . . 3 |
78 | 68, 74, 77 | sylancr 405 | . 2 |
79 | axltwlin 7180 | . . . . 5 | |
80 | 54, 79 | mp3an1 1255 | . . . 4 |
81 | 80 | ancoms 264 | . . 3 |
82 | 81 | 3impia 1135 | . 2 |
83 | 67, 78, 82 | mpjaodan 744 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 caddc 6984 cmul 6986 clt 7153 cle 7154 cmin 7279 cdiv 7760 cn 8039 cn0 8288 cexp 9475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: expnlbnd 9597 |
Copyright terms: Public domain | W3C validator |