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Mirrors > Home > ILE Home > Th. List > fzosplitprm1 | Unicode version |
Description: Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
Ref | Expression |
---|---|
fzosplitprm1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 998 |
. . . 4
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2 | simp2 999 |
. . . 4
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3 | zre 9271 |
. . . . . 6
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4 | zre 9271 |
. . . . . 6
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5 | ltle 8059 |
. . . . . 6
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6 | 3, 4, 5 | syl2an 289 |
. . . . 5
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7 | 6 | 3impia 1201 |
. . . 4
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8 | eluz2 9548 |
. . . 4
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9 | 1, 2, 7, 8 | syl3anbrc 1182 |
. . 3
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10 | fzosplitsn 10247 |
. . 3
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11 | 9, 10 | syl 14 |
. 2
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12 | zcn 9272 |
. . . . . . 7
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13 | ax-1cn 7918 |
. . . . . . 7
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14 | npcan 8180 |
. . . . . . . 8
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15 | 14 | eqcomd 2193 |
. . . . . . 7
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16 | 12, 13, 15 | sylancl 413 |
. . . . . 6
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17 | 16 | 3ad2ant2 1020 |
. . . . 5
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18 | 17 | oveq2d 5904 |
. . . 4
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19 | peano2zm 9305 |
. . . . . . 7
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20 | 19 | 3ad2ant2 1020 |
. . . . . 6
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21 | zltlem1 9324 |
. . . . . . 7
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22 | 21 | biimp3a 1355 |
. . . . . 6
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23 | eluz2 9548 |
. . . . . 6
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24 | 1, 20, 22, 23 | syl3anbrc 1182 |
. . . . 5
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25 | fzosplitsn 10247 |
. . . . 5
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26 | 24, 25 | syl 14 |
. . . 4
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27 | 18, 26 | eqtrd 2220 |
. . 3
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28 | 27 | uneq1d 3300 |
. 2
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29 | unass 3304 |
. . 3
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30 | df-pr 3611 |
. . . . . 6
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31 | 30 | eqcomi 2191 |
. . . . 5
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32 | 31 | a1i 9 |
. . . 4
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33 | 32 | uneq2d 3301 |
. . 3
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34 | 29, 33 | eqtrid 2232 |
. 2
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35 | 11, 28, 34 | 3eqtrd 2224 |
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 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-apti 7940 ax-pre-ltadd 7941 |
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-csb 3070 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-iun 3900 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-1st 6155 df-2nd 6156 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-n0 9191 df-z 9268 df-uz 9543 df-fz 10023 df-fzo 10157 |
This theorem is referenced by: (None) |
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