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Mirrors > Home > ILE Home > Th. List > hashfzp1 | Unicode version |
Description: Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
hashfzp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9550 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | eluzelz 9554 |
. . . 4
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3 | zdceq 9345 |
. . . 4
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4 | 1, 2, 3 | syl2anc 411 |
. . 3
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5 | exmiddc 837 |
. . 3
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6 | 4, 5 | syl 14 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | hash0 10793 |
. . . . 5
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8 | eluzelre 9555 |
. . . . . . . 8
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9 | 8 | ltp1d 8904 |
. . . . . . 7
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10 | peano2z 9306 |
. . . . . . . . 9
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11 | 10 | ancri 324 |
. . . . . . . 8
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12 | fzn 10059 |
. . . . . . . 8
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13 | 2, 11, 12 | 3syl 17 |
. . . . . . 7
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14 | 9, 13 | mpbid 147 |
. . . . . 6
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15 | 14 | fveq2d 5533 |
. . . . 5
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16 | 2 | zcnd 9393 |
. . . . . 6
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17 | 16 | subidd 8273 |
. . . . 5
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18 | 7, 15, 17 | 3eqtr4a 2247 |
. . . 4
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19 | oveq1 5897 |
. . . . . . 7
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20 | 19 | oveq1d 5905 |
. . . . . 6
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21 | 20 | fveq2d 5533 |
. . . . 5
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22 | oveq2 5898 |
. . . . 5
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23 | 21, 22 | eqeq12d 2203 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | imbitrrid 156 |
. . 3
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25 | uzp1 9578 |
. . . . . . . 8
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26 | pm2.24 622 |
. . . . . . . . . 10
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27 | 26 | eqcoms 2191 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | ax-1 6 |
. . . . . . . . 9
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29 | 27, 28 | jaoi 717 |
. . . . . . . 8
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30 | 25, 29 | syl 14 |
. . . . . . 7
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31 | 30 | impcom 125 |
. . . . . 6
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32 | hashfz 10818 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 31, 32 | syl 14 |
. . . . 5
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34 | 1 | zcnd 9393 |
. . . . . . 7
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35 | 1cnd 7990 |
. . . . . . 7
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36 | 16, 34, 35 | nppcan2d 8311 |
. . . . . 6
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37 | 36 | adantl 277 |
. . . . 5
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38 | 33, 37 | eqtrd 2221 |
. . . 4
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39 | 38 | ex 115 |
. . 3
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40 | 24, 39 | jaoi 717 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 6, 40 | mpcom 36 |
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 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-iord 4380 df-on 4382 df-ilim 4383 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-recs 6323 df-frec 6409 df-1o 6434 df-er 6552 df-en 6758 df-dom 6759 df-fin 6760 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-inn 8937 df-n0 9194 df-z 9271 df-uz 9546 df-fz 10026 df-ihash 10773 |
This theorem is referenced by: (None) |
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