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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 10318 | . . 3 ⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴))) | |
2 | intqfrac2.2 | . . . 4 ⊢ 𝐹 = (𝐴 − 𝑍) | |
3 | intqfrac2.1 | . . . . 5 ⊢ 𝑍 = (⌊‘𝐴) | |
4 | 3 | oveq2i 5911 | . . . 4 ⊢ (𝐴 − 𝑍) = (𝐴 − (⌊‘𝐴)) |
5 | 2, 4 | eqtri 2210 | . . 3 ⊢ 𝐹 = (𝐴 − (⌊‘𝐴)) |
6 | 1, 5 | breqtrrdi 4063 | . 2 ⊢ (𝐴 ∈ ℚ → 0 ≤ 𝐹) |
7 | qfraclt1 10317 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1) | |
8 | 5, 7 | eqbrtrid 4056 | . 2 ⊢ (𝐴 ∈ ℚ → 𝐹 < 1) |
9 | 2 | oveq2i 5911 | . . 3 ⊢ (𝑍 + 𝐹) = (𝑍 + (𝐴 − 𝑍)) |
10 | flqcl 10310 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | |
11 | 3, 10 | eqeltrid 2276 | . . . . 5 ⊢ (𝐴 ∈ ℚ → 𝑍 ∈ ℤ) |
12 | 11 | zcnd 9411 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝑍 ∈ ℂ) |
13 | qcn 9670 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
14 | 12, 13 | pncan3d 8306 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝑍 + (𝐴 − 𝑍)) = 𝐴) |
15 | 9, 14 | eqtr2id 2235 | . 2 ⊢ (𝐴 ∈ ℚ → 𝐴 = (𝑍 + 𝐹)) |
16 | 6, 8, 15 | 3jca 1179 | 1 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 ‘cfv 5238 (class class class)co 5900 0cc0 7846 1c1 7847 + caddc 7849 < clt 8027 ≤ cle 8028 − cmin 8163 ℤcz 9288 ℚcq 9655 ⌊cfl 10305 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-po 4317 df-iso 4318 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-n0 9212 df-z 9289 df-q 9656 df-rp 9690 df-fl 10307 |
This theorem is referenced by: intfracq 10357 flqdiv 10358 |
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