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Mirrors > Home > MPE Home > Th. List > fladdz | Structured version Visualization version GIF version |
Description: An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Ref | Expression |
---|---|
fladdz | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reflcl 13169 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ) |
3 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ) | |
4 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | 4 | zred 12090 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ) |
6 | flle 13172 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴) |
8 | 2, 3, 5, 7 | leadd1dd 11256 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁)) |
9 | 1red 10644 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ) | |
10 | 2, 9 | readdcld 10672 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ) |
11 | flltp1 13173 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1)) |
13 | 3, 10, 5, 12 | ltadd1dd 11253 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 1) + 𝑁)) |
14 | 2 | recnd 10671 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℂ) |
15 | 1cnd 10638 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℂ) | |
16 | 5 | recnd 10671 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ) |
17 | 14, 15, 16 | add32d 10869 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (((⌊‘𝐴) + 1) + 𝑁) = (((⌊‘𝐴) + 𝑁) + 1)) |
18 | 13, 17 | breqtrd 5094 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1)) |
19 | 3, 5 | readdcld 10672 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) ∈ ℝ) |
20 | 3 | flcld 13171 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ) |
21 | 20, 4 | zaddcld 12094 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ∈ ℤ) |
22 | flbi 13189 | . . 3 ⊢ (((𝐴 + 𝑁) ∈ ℝ ∧ ((⌊‘𝐴) + 𝑁) ∈ ℤ) → ((⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁) ↔ (((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁) ∧ (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1)))) | |
23 | 19, 21, 22 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁) ↔ (((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁) ∧ (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1)))) |
24 | 8, 18, 23 | mpbir2and 711 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 ℤcz 11984 ⌊cfl 13163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fl 13165 |
This theorem is referenced by: flzadd 13199 modcyc 13277 bitsmod 15787 fldivp1 16235 ppip1le 25740 dya2ub 31530 fourierdlem4 42403 fourierdlem47 42445 flsubz 44584 blennnt2 44656 |
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