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Mirrors > Home > ILE Home > Th. List > flqcld | GIF version |
Description: The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Ref | Expression |
---|---|
flqcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℚ) |
Ref | Expression |
---|---|
flqcld | ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flqcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℚ) | |
2 | flqcl 10202 | . 2 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ‘cfv 5185 ℤcz 9185 ℚcq 9551 ⌊cfl 10197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 ax-arch 7866 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-id 4268 df-po 4271 df-iso 4272 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 df-n0 9109 df-z 9186 df-q 9552 df-rp 9584 df-fl 10199 |
This theorem is referenced by: flqge 10211 flqlt 10212 flid 10213 flqltnz 10216 flqwordi 10217 flqword2 10218 flqaddz 10226 flhalf 10231 flltdivnn0lt 10233 fldiv4p1lem1div2 10234 ceiqcl 10236 ceiqge 10238 ceiqm1l 10240 intfracq 10249 flqdiv 10250 modqval 10253 modqvalr 10254 modqcl 10255 flqpmodeq 10256 modq0 10258 modqge0 10261 modqlt 10262 modqdiffl 10264 modqdifz 10265 modqmulnn 10271 modqvalp1 10272 zmodcl 10273 modqcyc 10288 modqadd1 10290 modqmuladd 10295 modqmul1 10306 modqdi 10321 modqsubdir 10322 iexpcyc 10553 facavg 10653 dvdsmod 11794 divalglemnn 11849 divalgmod 11858 flodddiv4t2lthalf 11868 modgcd 11918 hashdvds 12147 prmdiv 12161 odzdvds 12171 fldivp1 12272 pcfac 12274 pcbc 12275 |
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