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Mirrors > Home > ILE Home > Th. List > intqfrac2 | GIF version |
Description: Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
intqfrac2.1 | ⊢ 𝑍 = (⌊‘𝐴) |
intqfrac2.2 | ⊢ 𝐹 = (𝐴 − 𝑍) |
Ref | Expression |
---|---|
intqfrac2 | ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qfracge0 10047 | . . 3 ⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴))) | |
2 | intqfrac2.2 | . . . 4 ⊢ 𝐹 = (𝐴 − 𝑍) | |
3 | intqfrac2.1 | . . . . 5 ⊢ 𝑍 = (⌊‘𝐴) | |
4 | 3 | oveq2i 5778 | . . . 4 ⊢ (𝐴 − 𝑍) = (𝐴 − (⌊‘𝐴)) |
5 | 2, 4 | eqtri 2158 | . . 3 ⊢ 𝐹 = (𝐴 − (⌊‘𝐴)) |
6 | 1, 5 | breqtrrdi 3965 | . 2 ⊢ (𝐴 ∈ ℚ → 0 ≤ 𝐹) |
7 | qfraclt1 10046 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1) | |
8 | 5, 7 | eqbrtrid 3958 | . 2 ⊢ (𝐴 ∈ ℚ → 𝐹 < 1) |
9 | 2 | oveq2i 5778 | . . 3 ⊢ (𝑍 + 𝐹) = (𝑍 + (𝐴 − 𝑍)) |
10 | flqcl 10039 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | |
11 | 3, 10 | eqeltrid 2224 | . . . . 5 ⊢ (𝐴 ∈ ℚ → 𝑍 ∈ ℤ) |
12 | 11 | zcnd 9167 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝑍 ∈ ℂ) |
13 | qcn 9419 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
14 | 12, 13 | pncan3d 8069 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝑍 + (𝐴 − 𝑍)) = 𝐴) |
15 | 9, 14 | syl5req 2183 | . 2 ⊢ (𝐴 ∈ ℚ → 𝐴 = (𝑍 + 𝐹)) |
16 | 6, 8, 15 | 3jca 1161 | 1 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 < clt 7793 ≤ cle 7794 − cmin 7926 ℤcz 9047 ℚcq 9404 ⌊cfl 10034 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-q 9405 df-rp 9435 df-fl 10036 |
This theorem is referenced by: intfracq 10086 flqdiv 10087 |
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