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Mirrors > Home > MPE Home > Th. List > root1eq1 | Structured version Visualization version Unicode version |
Description: The only powers of an
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Ref | Expression |
---|---|
root1eq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 10686 |
. . . . . . . 8
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2 | simpl 459 |
. . . . . . . 8
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3 | nndivre 10652 |
. . . . . . . 8
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4 | 1, 2, 3 | sylancr 670 |
. . . . . . 7
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5 | 4 | recnd 9674 |
. . . . . 6
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6 | ax-icn 9603 |
. . . . . . . 8
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7 | picn 23426 |
. . . . . . . 8
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8 | 6, 7 | mulcli 9653 |
. . . . . . 7
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9 | 8 | a1i 11 |
. . . . . 6
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10 | 5, 9 | mulcld 9668 |
. . . . 5
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11 | efexp 14167 |
. . . . 5
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12 | 10, 11 | sylancom 674 |
. . . 4
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13 | zcn 10949 |
. . . . . . . . 9
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14 | 13 | adantl 468 |
. . . . . . . 8
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15 | nncn 10624 |
. . . . . . . . 9
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16 | 15 | adantr 467 |
. . . . . . . 8
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17 | 2cn 10687 |
. . . . . . . . 9
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18 | 17 | a1i 11 |
. . . . . . . 8
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19 | nnne0 10649 |
. . . . . . . . 9
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20 | 19 | adantr 467 |
. . . . . . . 8
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21 | 14, 16, 18, 20 | div32d 10413 |
. . . . . . 7
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22 | 21 | oveq1d 6310 |
. . . . . 6
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23 | 14, 16, 20 | divcld 10390 |
. . . . . . 7
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24 | 23, 18, 9 | mulassd 9671 |
. . . . . 6
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25 | 14, 5, 9 | mulassd 9671 |
. . . . . 6
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26 | 22, 24, 25 | 3eqtr3d 2495 |
. . . . 5
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27 | 26 | fveq2d 5874 |
. . . 4
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28 | neg1cn 10720 |
. . . . . . . 8
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29 | 28 | a1i 11 |
. . . . . . 7
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30 | neg1ne0 10722 |
. . . . . . . 8
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31 | 30 | a1i 11 |
. . . . . . 7
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32 | 29, 31, 5 | cxpefd 23669 |
. . . . . 6
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33 | logm1 23550 |
. . . . . . . 8
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34 | 33 | oveq2i 6306 |
. . . . . . 7
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35 | 34 | fveq2i 5873 |
. . . . . 6
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36 | 32, 35 | syl6eq 2503 |
. . . . 5
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37 | 36 | oveq1d 6310 |
. . . 4
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38 | 12, 27, 37 | 3eqtr4rd 2498 |
. . 3
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39 | 38 | eqeq1d 2455 |
. 2
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40 | 17, 8 | mulcli 9653 |
. . . 4
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41 | mulcl 9628 |
. . . 4
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42 | 23, 40, 41 | sylancl 669 |
. . 3
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43 | efeq1 23490 |
. . 3
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44 | 42, 43 | syl 17 |
. 2
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45 | 6, 17, 7 | mul12i 9833 |
. . . . . 6
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46 | 45 | oveq2i 6306 |
. . . . 5
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47 | 40 | a1i 11 |
. . . . . 6
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48 | 2ne0 10709 |
. . . . . . . 8
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49 | ine0 10061 |
. . . . . . . . 9
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50 | pire 23425 |
. . . . . . . . . 10
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51 | pipos 23427 |
. . . . . . . . . 10
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52 | 50, 51 | gt0ne0ii 10157 |
. . . . . . . . 9
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53 | 6, 7, 49, 52 | mulne0i 10262 |
. . . . . . . 8
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54 | 17, 8, 48, 53 | mulne0i 10262 |
. . . . . . 7
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55 | 54 | a1i 11 |
. . . . . 6
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56 | 23, 47, 55 | divcan4d 10396 |
. . . . 5
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57 | 46, 56 | syl5eq 2499 |
. . . 4
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58 | 57 | eleq1d 2515 |
. . 3
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59 | nnz 10966 |
. . . . 5
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60 | 59 | adantr 467 |
. . . 4
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61 | simpr 463 |
. . . 4
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62 | dvdsval2 14320 |
. . . 4
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63 | 60, 20, 61, 62 | syl3anc 1269 |
. . 3
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64 | 58, 63 | bitr4d 260 |
. 2
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65 | 39, 44, 64 | 3bitrd 283 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-inf2 8151 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 ax-pre-sup 9622 ax-addf 9623 ax-mulf 9624 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-fal 1452 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-se 4797 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-isom 5594 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-of 6536 df-om 6698 df-1st 6798 df-2nd 6799 df-supp 6920 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-2o 7188 df-oadd 7191 df-er 7368 df-map 7479 df-pm 7480 df-ixp 7528 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-fsupp 7889 df-fi 7930 df-sup 7961 df-inf 7962 df-oi 8030 df-card 8378 df-cda 8603 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-div 10277 df-nn 10617 df-2 10675 df-3 10676 df-4 10677 df-5 10678 df-6 10679 df-7 10680 df-8 10681 df-9 10682 df-10 10683 df-n0 10877 df-z 10945 df-dec 11059 df-uz 11167 df-q 11272 df-rp 11310 df-xneg 11416 df-xadd 11417 df-xmul 11418 df-ioo 11646 df-ioc 11647 df-ico 11648 df-icc 11649 df-fz 11792 df-fzo 11923 df-fl 12035 df-mod 12104 df-seq 12221 df-exp 12280 df-fac 12467 df-bc 12495 df-hash 12523 df-shft 13142 df-cj 13174 df-re 13175 df-im 13176 df-sqrt 13310 df-abs 13311 df-limsup 13538 df-clim 13564 df-rlim 13565 df-sum 13765 df-ef 14133 df-sin 14135 df-cos 14136 df-pi 14138 df-dvds 14318 df-struct 15135 df-ndx 15136 df-slot 15137 df-base 15138 df-sets 15139 df-ress 15140 df-plusg 15215 df-mulr 15216 df-starv 15217 df-sca 15218 df-vsca 15219 df-ip 15220 df-tset 15221 df-ple 15222 df-ds 15224 df-unif 15225 df-hom 15226 df-cco 15227 df-rest 15333 df-topn 15334 df-0g 15352 df-gsum 15353 df-topgen 15354 df-pt 15355 df-prds 15358 df-xrs 15412 df-qtop 15418 df-imas 15419 df-xps 15422 df-mre 15504 df-mrc 15505 df-acs 15507 df-mgm 16500 df-sgrp 16539 df-mnd 16549 df-submnd 16595 df-mulg 16688 df-cntz 16983 df-cmn 17444 df-psmet 18974 df-xmet 18975 df-met 18976 df-bl 18977 df-mopn 18978 df-fbas 18979 df-fg 18980 df-cnfld 18983 df-top 19933 df-bases 19934 df-topon 19935 df-topsp 19936 df-cld 20046 df-ntr 20047 df-cls 20048 df-nei 20126 df-lp 20164 df-perf 20165 df-cn 20255 df-cnp 20256 df-haus 20343 df-tx 20589 df-hmeo 20782 df-fil 20873 df-fm 20965 df-flim 20966 df-flf 20967 df-xms 21347 df-ms 21348 df-tms 21349 df-cncf 21922 df-limc 22833 df-dv 22834 df-log 23518 df-cxp 23519 |
This theorem is referenced by: dchrptlem1 24204 dchrptlem2 24205 |
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