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Mirrors > Home > ILE Home > Th. List > intqfrac | GIF version |
Description: Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
intqfrac | ⊢ (𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqfrac 9471 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴))) | |
2 | 1 | oveq2d 5579 | . 2 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) + (𝐴 mod 1)) = ((⌊‘𝐴) + (𝐴 − (⌊‘𝐴)))) |
3 | flqcl 9407 | . . . 4 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | |
4 | 3 | zcnd 8603 | . . 3 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℂ) |
5 | qcn 8852 | . . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
6 | 4, 5 | pncan3d 7541 | . 2 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) + (𝐴 − (⌊‘𝐴))) = 𝐴) |
7 | 2, 6 | eqtr2d 2116 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ‘cfv 4952 (class class class)co 5563 1c1 7096 + caddc 7098 − cmin 7398 ℚcq 8837 ⌊cfl 9402 mod cmo 9456 |
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 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 ax-pre-mulext 7208 ax-arch 7209 |
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 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-id 4076 df-po 4079 df-iso 4080 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-pnf 7269 df-mnf 7270 df-xr 7271 df-ltxr 7272 df-le 7273 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 df-div 7880 df-inn 8159 df-n0 8408 df-z 8485 df-q 8838 df-rp 8868 df-fl 9404 df-mod 9457 |
This theorem is referenced by: (None) |
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