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Mirrors > Home > ILE Home > Th. List > lt2msq | Unicode version |
Description: Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
lt2msq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2msq1 8831 |
. . . 4
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2 | 1 | 3expia 1205 |
. . 3
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3 | 2 | adantrr 479 |
. 2
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4 | simpr 110 |
. . . . . . 7
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5 | simpll 527 |
. . . . . . 7
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6 | lt2msq1 8831 |
. . . . . . . 8
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7 | 6 | 3expia 1205 |
. . . . . . 7
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8 | 4, 5, 7 | syl2anc 411 |
. . . . . 6
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9 | 8 | con3d 631 |
. . . . 5
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10 | 5, 5 | remulcld 7978 |
. . . . . 6
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11 | simprl 529 |
. . . . . . 7
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12 | 11, 11 | remulcld 7978 |
. . . . . 6
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13 | 10, 12 | lenltd 8065 |
. . . . 5
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14 | 5, 11 | lenltd 8065 |
. . . . 5
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15 | 9, 13, 14 | 3imtr4d 203 |
. . . 4
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16 | 5 | recnd 7976 |
. . . . . 6
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17 | 11 | recnd 7976 |
. . . . . 6
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18 | mulext 8561 |
. . . . . 6
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19 | 16, 16, 17, 17, 18 | syl22anc 1239 |
. . . . 5
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20 | oridm 757 |
. . . . 5
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21 | 19, 20 | syl6ib 161 |
. . . 4
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22 | 15, 21 | anim12d 335 |
. . 3
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23 | ltleap 8579 |
. . . 4
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24 | 10, 12, 23 | syl2anc 411 |
. . 3
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25 | ltleap 8579 |
. . . 4
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26 | 5, 11, 25 | syl2anc 411 |
. . 3
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27 | 22, 24, 26 | 3imtr4d 203 |
. 2
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28 | 3, 27 | impbid 129 |
1
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 |
This theorem is referenced by: le2msq 8847 lt2msqi 8860 lt2sq 10579 |
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