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Mirrors > Home > MPE Home > Th. List > intfrac2 | Structured version Visualization version GIF version |
Description: Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12547? (Contributed by NM, 16-Aug-2008.) |
Ref | Expression |
---|---|
intfrac2.1 | ⊢ 𝑍 = (⌊‘𝐴) |
intfrac2.2 | ⊢ 𝐹 = (𝐴 − 𝑍) |
Ref | Expression |
---|---|
intfrac2 | ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fracge0 12467 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | |
2 | intfrac2.2 | . . . 4 ⊢ 𝐹 = (𝐴 − 𝑍) | |
3 | intfrac2.1 | . . . . 5 ⊢ 𝑍 = (⌊‘𝐴) | |
4 | 3 | oveq2i 6560 | . . . 4 ⊢ (𝐴 − 𝑍) = (𝐴 − (⌊‘𝐴)) |
5 | 2, 4 | eqtri 2632 | . . 3 ⊢ 𝐹 = (𝐴 − (⌊‘𝐴)) |
6 | 1, 5 | syl6breqr 4625 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ 𝐹) |
7 | fraclt1 12465 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | |
8 | 5, 7 | syl5eqbr 4618 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐹 < 1) |
9 | 2 | oveq2i 6560 | . . 3 ⊢ (𝑍 + 𝐹) = (𝑍 + (𝐴 − 𝑍)) |
10 | flcl 12458 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
11 | 3, 10 | syl5eqel 2692 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝑍 ∈ ℤ) |
12 | 11 | zcnd 11359 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝑍 ∈ ℂ) |
13 | recn 9905 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | 12, 13 | pncan3d 10274 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝑍 + (𝐴 − 𝑍)) = 𝐴) |
15 | 9, 14 | syl5req 2657 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 = (𝑍 + 𝐹)) |
16 | 6, 8, 15 | 3jca 1235 | 1 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ⌊cfl 12453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fl 12455 |
This theorem is referenced by: intfracq 12520 fldiv 12521 |
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