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Mirrors > Home > ILE Home > Th. List > fzrev | Unicode version |
Description: Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Ref | Expression |
---|---|
fzrev |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9246 |
. . . . . . . 8
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2 | zre 9246 |
. . . . . . . 8
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3 | zre 9246 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | suble 8387 |
. . . . . . . 8
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5 | 1, 2, 3, 4 | syl3an 1280 |
. . . . . . 7
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6 | 5 | 3comr 1211 |
. . . . . 6
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7 | 6 | 3expb 1204 |
. . . . 5
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8 | 7 | adantll 476 |
. . . 4
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9 | zre 9246 |
. . . . . . 7
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10 | lesub 8388 |
. . . . . . 7
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11 | 9, 1, 2, 10 | syl3an 1280 |
. . . . . 6
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12 | 11 | 3expb 1204 |
. . . . 5
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13 | 12 | adantlr 477 |
. . . 4
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14 | 8, 13 | anbi12d 473 |
. . 3
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15 | ancom 266 |
. . 3
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16 | 14, 15 | bitr3di 195 |
. 2
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17 | simprr 531 |
. . 3
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18 | zsubcl 9283 |
. . . . 5
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19 | 18 | ancoms 268 |
. . . 4
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20 | 19 | ad2ant2lr 510 |
. . 3
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21 | zsubcl 9283 |
. . . . 5
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22 | 21 | ancoms 268 |
. . . 4
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23 | 22 | ad2ant2r 509 |
. . 3
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24 | elfz 10001 |
. . 3
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25 | 17, 20, 23, 24 | syl3anc 1238 |
. 2
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26 | zsubcl 9283 |
. . . 4
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27 | 26 | adantl 277 |
. . 3
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28 | simpll 527 |
. . 3
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29 | simplr 528 |
. . 3
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30 | elfz 10001 |
. . 3
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31 | 27, 28, 29, 30 | syl3anc 1238 |
. 2
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32 | 16, 25, 31 | 3bitr4d 220 |
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-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-int 3843 df-br 4001 df-opab 4062 df-id 4290 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-inn 8909 df-n0 9166 df-z 9243 df-fz 9996 |
This theorem is referenced by: fzrev2 10071 fzrev3 10073 fzrevral 10091 fsumrev 11435 fprodrev 11611 |
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