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Mirrors > Home > MPE Home > Th. List > asinlem | Structured version Visualization version Unicode version |
Description: The argument to the logarithm in df-asin 23839 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
asinlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 9623 |
. . . 4
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2 | mulcl 9648 |
. . . 4
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3 | 1, 2 | mpan 681 |
. . 3
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4 | ax-1cn 9622 |
. . . . 5
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5 | sqcl 12368 |
. . . . 5
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6 | subcl 9899 |
. . . . 5
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7 | 4, 5, 6 | sylancr 674 |
. . . 4
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8 | 7 | sqrtcld 13547 |
. . 3
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9 | 3, 8 | subnegd 10018 |
. 2
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10 | 8 | negcld 9998 |
. . 3
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11 | 0ne1 10704 |
. . . . . 6
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12 | 0cnd 9661 |
. . . . . . 7
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13 | 1cnd 9684 |
. . . . . . 7
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14 | subcan2 9924 |
. . . . . . . 8
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15 | 14 | necon3bid 2679 |
. . . . . . 7
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16 | 12, 13, 5, 15 | syl3anc 1276 |
. . . . . 6
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17 | 11, 16 | mpbiri 241 |
. . . . 5
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18 | sqmul 12369 |
. . . . . . . 8
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19 | 1, 18 | mpan 681 |
. . . . . . 7
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20 | i2 12406 |
. . . . . . . . 9
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21 | 20 | oveq1i 6324 |
. . . . . . . 8
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22 | 5 | mulm1d 10097 |
. . . . . . . 8
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23 | 21, 22 | syl5eq 2507 |
. . . . . . 7
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24 | 19, 23 | eqtrd 2495 |
. . . . . 6
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25 | df-neg 9888 |
. . . . . 6
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26 | 24, 25 | syl6eq 2511 |
. . . . 5
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27 | sqneg 12366 |
. . . . . . 7
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28 | 8, 27 | syl 17 |
. . . . . 6
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29 | 7 | sqsqrtd 13549 |
. . . . . 6
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30 | 28, 29 | eqtrd 2495 |
. . . . 5
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31 | 17, 26, 30 | 3netr4d 2712 |
. . . 4
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32 | oveq1 6321 |
. . . . 5
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33 | 32 | necon3i 2667 |
. . . 4
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34 | 31, 33 | syl 17 |
. . 3
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35 | 3, 10, 34 | subne0d 10020 |
. 2
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36 | 9, 35 | eqnetrrd 2703 |
1
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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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-pre-sup 9642 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-om 6719 df-2nd 6820 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-er 7388 df-en 7595 df-dom 7596 df-sdom 7597 df-sup 7981 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-div 10297 df-nn 10637 df-2 10695 df-3 10696 df-n0 10898 df-z 10966 df-uz 11188 df-rp 11331 df-seq 12245 df-exp 12304 df-cj 13210 df-re 13211 df-im 13212 df-sqrt 13346 df-abs 13347 |
This theorem is referenced by: asinlem3 23845 asinf 23846 asinneg 23860 efiasin 23862 asinbnd 23873 dvasin 32072 |
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