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Mirrors > Home > ILE Home > Th. List > rebtwn2z | Unicode version |
Description: A real number can be
bounded by integers above and below which are two
apart.
The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Ref | Expression |
---|---|
rebtwn2z |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnz 9386 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | reeanv 2657 |
. . 3
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3 | 1, 2 | sylibr 134 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | simpll 527 |
. . . . 5
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5 | simplrl 535 |
. . . . . . . . 9
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6 | 5 | zred 9389 |
. . . . . . . 8
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7 | simplrr 536 |
. . . . . . . . 9
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8 | 7 | zred 9389 |
. . . . . . . 8
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9 | simprl 529 |
. . . . . . . 8
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10 | simprr 531 |
. . . . . . . 8
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11 | 6, 4, 8, 9, 10 | lttrd 8097 |
. . . . . . 7
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12 | znnsub 9318 |
. . . . . . . 8
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13 | 12 | ad2antlr 489 |
. . . . . . 7
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14 | 11, 13 | mpbid 147 |
. . . . . 6
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15 | elnnuz 9578 |
. . . . . . . 8
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16 | eluzp1p1 9567 |
. . . . . . . 8
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17 | 15, 16 | sylbi 121 |
. . . . . . 7
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18 | df-2 8992 |
. . . . . . . 8
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19 | 18 | fveq2i 5530 |
. . . . . . 7
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20 | 17, 19 | eleqtrrdi 2281 |
. . . . . 6
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21 | 14, 20 | syl 14 |
. . . . 5
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22 | 5 | zcnd 9390 |
. . . . . . . . 9
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23 | 7 | zcnd 9390 |
. . . . . . . . 9
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24 | 22, 23 | pncan3d 8285 |
. . . . . . . 8
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25 | 24, 8 | eqeltrd 2264 |
. . . . . . 7
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26 | 8, 6 | resubcld 8352 |
. . . . . . . . 9
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27 | 1red 7986 |
. . . . . . . . 9
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28 | 26, 27 | readdcld 8001 |
. . . . . . . 8
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29 | 6, 28 | readdcld 8001 |
. . . . . . 7
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30 | 10, 24 | breqtrrd 4043 |
. . . . . . 7
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31 | 26 | ltp1d 8901 |
. . . . . . . 8
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32 | 26, 28, 6, 31 | ltadd2dd 8393 |
. . . . . . 7
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33 | 4, 25, 29, 30, 32 | lttrd 8097 |
. . . . . 6
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34 | breq1 4018 |
. . . . . . . 8
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35 | oveq1 5895 |
. . . . . . . . 9
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36 | 35 | breq2d 4027 |
. . . . . . . 8
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37 | 34, 36 | anbi12d 473 |
. . . . . . 7
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38 | 37 | rspcev 2853 |
. . . . . 6
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39 | 5, 9, 33, 38 | syl12anc 1246 |
. . . . 5
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40 | rebtwn2zlemshrink 10268 |
. . . . 5
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41 | 4, 21, 39, 40 | syl3anc 1248 |
. . . 4
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42 | 41 | ex 115 |
. . 3
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43 | 42 | rexlimdvva 2612 |
. 2
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44 | 3, 43 | mpd 13 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-ltadd 7941 ax-arch 7944 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-inn 8934 df-2 8992 df-n0 9191 df-z 9268 df-uz 9543 |
This theorem is referenced by: qbtwnre 10271 |
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