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Mirrors > Home > ILE Home > Th. List > hashfz | Unicode version |
Description: Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.) |
Ref | Expression |
---|---|
hashfz |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9568 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | eluzelz 9572 |
. . . . 5
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3 | 1z 9314 |
. . . . . 6
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4 | zsubcl 9329 |
. . . . . 6
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5 | 3, 1, 4 | sylancr 414 |
. . . . 5
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6 | fzen 10079 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 5, 6 | syl3anc 1249 |
. . . 4
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8 | 1 | zcnd 9411 |
. . . . . 6
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9 | ax-1cn 7939 |
. . . . . 6
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10 | pncan3 8200 |
. . . . . 6
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11 | 8, 9, 10 | sylancl 413 |
. . . . 5
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12 | 1cnd 8008 |
. . . . . 6
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13 | 2 | zcnd 9411 |
. . . . . . 7
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14 | 13, 8 | subcld 8303 |
. . . . . 6
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15 | 13, 12, 8 | addsub12d 8326 |
. . . . . 6
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16 | 12, 14, 15 | comraddd 8149 |
. . . . 5
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17 | 11, 16 | oveq12d 5918 |
. . . 4
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18 | 7, 17 | breqtrd 4047 |
. . 3
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19 | 1, 2 | fzfigd 10468 |
. . . 4
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20 | 1zzd 9315 |
. . . . 5
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21 | 2, 1 | zsubcld 9415 |
. . . . . 6
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22 | 21 | peano2zd 9413 |
. . . . 5
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23 | 20, 22 | fzfigd 10468 |
. . . 4
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24 | hashen 10805 |
. . . 4
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25 | 19, 23, 24 | syl2anc 411 |
. . 3
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26 | 18, 25 | mpbird 167 |
. 2
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27 | uznn0sub 9595 |
. . 3
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28 | peano2nn0 9251 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | hashfz1 10804 |
. . 3
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30 | 27, 28, 29 | 3syl 17 |
. 2
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31 | 26, 30 | eqtrd 2222 |
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-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 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-apti 7961 ax-pre-ltadd 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-1o 6445 df-er 6563 df-en 6771 df-dom 6772 df-fin 6773 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-uz 9564 df-fz 10045 df-ihash 10797 |
This theorem is referenced by: hashfzo 10843 hashfzp1 10845 hashfz0 10846 |
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