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Mirrors > Home > ILE Home > Th. List > fldiv4lem1div2 | Unicode version |
Description: The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
fldiv4lem1div2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn1uz2 9662 |
. 2
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2 | 1lt4 9146 |
. . . . . 6
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3 | 1nn0 9246 |
. . . . . . 7
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4 | 4nn 9135 |
. . . . . . 7
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5 | divfl0 10355 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 3, 4, 5 | mp2an 426 |
. . . . . 6
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7 | 2, 6 | mpbi 145 |
. . . . 5
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8 | 1z 9333 |
. . . . . . . . 9
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9 | znq 9679 |
. . . . . . . . 9
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10 | 8, 4, 9 | mp2an 426 |
. . . . . . . 8
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11 | flqcl 10332 |
. . . . . . . 8
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12 | 10, 11 | ax-mp 5 |
. . . . . . 7
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13 | 12 | zrei 9313 |
. . . . . 6
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14 | 13 | eqlei 8103 |
. . . . 5
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15 | 7, 14 | mp1i 10 |
. . . 4
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16 | fvoveq1 5933 |
. . . 4
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17 | oveq1 5917 |
. . . . . . 7
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18 | 1m1e0 9041 |
. . . . . . 7
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19 | 17, 18 | eqtrdi 2242 |
. . . . . 6
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20 | 19 | oveq1d 5925 |
. . . . 5
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21 | 2cn 9043 |
. . . . . 6
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22 | 2ap0 9065 |
. . . . . 6
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23 | 21, 22 | div0api 8755 |
. . . . 5
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24 | 20, 23 | eqtrdi 2242 |
. . . 4
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25 | 15, 16, 24 | 3brtr4d 4061 |
. . 3
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26 | fldiv4lem1div2uz2 10365 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | jaoi 717 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 1, 27 | sylbi 121 |
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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-po 4325 df-iso 4326 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-fl 10329 |
This theorem is referenced by: gausslemma2dlem0g 15113 |
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