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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 9567 | . 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 1434 ‘cfv 4967 ℤcz 8644 ℚcq 8997 ⌊cfl 9562 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-po 4086 df-iso 4087 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-n0 8564 df-z 8645 df-q 8998 df-rp 9028 df-fl 9564 |
This theorem is referenced by: flqge 9576 flqlt 9577 flid 9578 flqltnz 9581 flqwordi 9582 flqword2 9583 flqaddz 9591 flhalf 9596 flltdivnn0lt 9598 fldiv4p1lem1div2 9599 ceiqcl 9601 ceiqge 9603 ceiqm1l 9605 intfracq 9614 flqdiv 9615 modqval 9618 modqvalr 9619 modqcl 9620 flqpmodeq 9621 modq0 9623 modqge0 9626 modqlt 9627 modqdiffl 9629 modqdifz 9630 modqmulnn 9636 modqvalp1 9637 zmodcl 9638 modqcyc 9653 modqadd1 9655 modqmuladd 9660 modqmul1 9671 modqdi 9686 modqsubdir 9687 iexpcyc 9893 facavg 9987 dvdsmod 10641 divalglemnn 10696 divalgmod 10705 flodddiv4t2lthalf 10715 modgcd 10760 hashdvds 10975 |
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