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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 9292 |
. . . . . . . 8
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2 | zre 9292 |
. . . . . . . 8
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3 | zre 9292 |
. . . . . . . 8
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4 | suble 8432 |
. . . . . . . 8
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5 | 1, 2, 3, 4 | syl3an 1291 |
. . . . . . 7
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6 | 5 | 3comr 1213 |
. . . . . 6
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7 | 6 | 3expb 1206 |
. . . . 5
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8 | 7 | adantll 476 |
. . . 4
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9 | zre 9292 |
. . . . . . 7
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10 | lesub 8433 |
. . . . . . 7
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11 | 9, 1, 2, 10 | syl3an 1291 |
. . . . . 6
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12 | 11 | 3expb 1206 |
. . . . 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 9329 |
. . . . 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 9329 |
. . . . 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 10050 |
. . 3
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25 | 17, 20, 23, 24 | syl3anc 1249 |
. 2
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26 | zsubcl 9329 |
. . . 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 10050 |
. . 3
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31 | 27, 28, 29, 30 | syl3anc 1249 |
. 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-ltadd 7962 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-inn 8955 df-n0 9212 df-z 9289 df-fz 10045 |
This theorem is referenced by: fzrev2 10121 fzrev3 10123 fzrevral 10141 fsumrev 11492 fprodrev 11668 |
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