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Mirrors > Home > ILE Home > Th. List > fzoval | Unicode version |
Description: Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
fzoval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel1 10118 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | elfzel1 9997 |
. . . 4
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4 | 3 | a1i 9 |
. . 3
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5 | peano2zm 9267 |
. . . . . . 7
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6 | fzf 9986 |
. . . . . . . 8
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7 | 6 | fovcl 5973 |
. . . . . . 7
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8 | 5, 7 | sylan2 286 |
. . . . . 6
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9 | id 19 |
. . . . . . . 8
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10 | oveq1 5875 |
. . . . . . . 8
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11 | 9, 10 | oveqan12d 5887 |
. . . . . . 7
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12 | df-fzo 10116 |
. . . . . . 7
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13 | 11, 12 | ovmpoga 5997 |
. . . . . 6
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14 | 8, 13 | mpd3an3 1338 |
. . . . 5
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15 | 14 | eleq2d 2247 |
. . . 4
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16 | 15 | expcom 116 |
. . 3
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17 | 2, 4, 16 | pm5.21ndd 705 |
. 2
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18 | 17 | eqrdv 2175 |
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 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
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-csb 3058 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-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 df-fz 9983 df-fzo 10116 |
This theorem is referenced by: elfzo 10122 fzodcel 10125 fzon 10139 fzoss1 10144 fzoss2 10145 fzval3 10177 fzo0to2pr 10191 fzo0to3tp 10192 fzo0to42pr 10193 fzoend 10195 fzofzp1b 10201 elfzom1b 10202 peano2fzor 10205 fzoshftral 10211 zmodfzo 10320 zmodidfzo 10326 fzofig 10405 hashfzo 10773 fzosump1 11396 telfsumo 11445 fsumparts 11449 geoserap 11486 geo2sum2 11494 dfphi2 12190 reumodprminv 12223 |
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