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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lt2addrd | Unicode version |
Description: If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
Ref | Expression |
---|---|
lt2addrd.1 |
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lt2addrd.2 |
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lt2addrd.3 |
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lt2addrd.4 |
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Ref | Expression |
---|---|
lt2addrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2addrd.2 |
. . 3
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2 | lt2addrd.3 |
. . . . . 6
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3 | 1, 2 | readdcld 9071 |
. . . . 5
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4 | lt2addrd.1 |
. . . . 5
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5 | 3, 4 | resubcld 9421 |
. . . 4
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6 | 5 | rehalfcld 10170 |
. . 3
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7 | 1, 6 | resubcld 9421 |
. 2
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8 | 2, 6 | resubcld 9421 |
. 2
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9 | 2 | recnd 9070 |
. . . . . 6
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10 | 1 | recnd 9070 |
. . . . . . . . 9
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11 | 10, 9 | addcld 9063 |
. . . . . . . 8
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12 | 4 | recnd 9070 |
. . . . . . . 8
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13 | 11, 12 | subcld 9367 |
. . . . . . 7
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14 | 13 | halfcld 10168 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 9, 14, 14 | subsub4d 9398 |
. . . . 5
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16 | 15 | oveq2d 6056 |
. . . 4
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17 | 9, 14 | subcld 9367 |
. . . . 5
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18 | 10, 14, 17 | subadd23d 9389 |
. . . 4
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19 | 13 | 2halvesd 10169 |
. . . . . 6
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20 | 19, 13 | eqeltrd 2478 |
. . . . 5
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21 | 10, 9, 20 | addsubassd 9387 |
. . . 4
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22 | 16, 18, 21 | 3eqtr4d 2446 |
. . 3
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23 | 19 | oveq2d 6056 |
. . 3
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24 | 11, 12 | nncand 9372 |
. . 3
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25 | 22, 23, 24 | 3eqtrrd 2441 |
. 2
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26 | lt2addrd.4 |
. . . . 5
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27 | difrp 10601 |
. . . . . 6
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28 | 4, 3, 27 | syl2anc 643 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 26, 28 | mpbid 202 |
. . . 4
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30 | 29 | rphalfcld 10616 |
. . 3
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31 | 1, 30 | ltsubrpd 10632 |
. 2
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32 | 2, 30 | ltsubrpd 10632 |
. 2
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33 | oveq1 6047 |
. . . . 5
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34 | 33 | eqeq2d 2415 |
. . . 4
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35 | breq1 4175 |
. . . 4
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36 | 34, 35 | 3anbi12d 1255 |
. . 3
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37 | oveq2 6048 |
. . . . 5
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38 | 37 | eqeq2d 2415 |
. . . 4
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39 | breq1 4175 |
. . . 4
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40 | 38, 39 | 3anbi13d 1256 |
. . 3
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41 | 36, 40 | rspc2ev 3020 |
. 2
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42 | 7, 8, 25, 31, 32, 41 | syl113anc 1196 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: xlt2addrd 24077 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-po 4463 df-so 4464 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-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-riota 6508 df-er 6864 df-en 7069 df-dom 7070 df-sdom 7071 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-div 9634 df-2 10014 df-rp 10569 |
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