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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 13923? (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 13841 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | |
2 | intfrac2.2 | . . . 4 ⊢ 𝐹 = (𝐴 − 𝑍) | |
3 | intfrac2.1 | . . . . 5 ⊢ 𝑍 = (⌊‘𝐴) | |
4 | 3 | oveq2i 7442 | . . . 4 ⊢ (𝐴 − 𝑍) = (𝐴 − (⌊‘𝐴)) |
5 | 2, 4 | eqtri 2763 | . . 3 ⊢ 𝐹 = (𝐴 − (⌊‘𝐴)) |
6 | 1, 5 | breqtrrdi 5190 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ 𝐹) |
7 | fraclt1 13839 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | |
8 | 5, 7 | eqbrtrid 5183 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐹 < 1) |
9 | 2 | oveq2i 7442 | . . 3 ⊢ (𝑍 + 𝐹) = (𝑍 + (𝐴 − 𝑍)) |
10 | flcl 13832 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
11 | 3, 10 | eqeltrid 2843 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝑍 ∈ ℤ) |
12 | 11 | zcnd 12721 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝑍 ∈ ℂ) |
13 | recn 11243 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | 12, 13 | pncan3d 11621 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝑍 + (𝐴 − 𝑍)) = 𝐴) |
15 | 9, 14 | eqtr2id 2788 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 = (𝑍 + 𝐹)) |
16 | 6, 8, 15 | 3jca 1127 | 1 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℤcz 12611 ⌊cfl 13827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fl 13829 |
This theorem is referenced by: intfracq 13896 fldiv 13897 |
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