fladdz - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fladdz		Structured version Visualization version GIF version

Theorem fladdz 13190

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.)

**Assertion**
Ref	Expression
fladdz	⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

**Proof of Theorem fladdz**
Step	Hyp	Ref	Expression
1		reflcl 13161	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
2	1	adantr 484	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ)
3		simpl 486	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ)
4		simpr 488	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
5	4	zred 12075	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		flle 13164	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
7	6	adantr 484	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴)
8	2, 3, 5, 7	leadd1dd 11243	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁))
9		1red 10631	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
10	2, 9	readdcld 10659	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ)
11		flltp1 13165	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
12	11	adantr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1))
13	3, 10, 5, 12	ltadd1dd 11240	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 1) + 𝑁))
14	2	recnd 10658	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℂ)
15		1cnd 10625	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℂ)
16	5	recnd 10658	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
17	14, 15, 16	add32d 10856	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (((⌊‘𝐴) + 1) + 𝑁) = (((⌊‘𝐴) + 𝑁) + 1))
18	13, 17	breqtrd 5056	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))
19	3, 5	readdcld 10659	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) ∈ ℝ)
20	3	flcld 13163	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ)
21	20, 4	zaddcld 12079	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ∈ ℤ)
22		flbi 13181	. . 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 587	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁) ↔ (((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁) ∧ (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))))
24	8, 18, 23	mpbir2and 712	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℤcz 11969 ⌊cfl 13155

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fl 13157

This theorem is referenced by: flzadd 13191 modcyc 13269 bitsmod 15775 fldivp1 16223 ppip1le 25746 dya2ub 31638 fourierdlem4 42753 fourierdlem47 42795 flsubz 44931 blennnt2 45003

W3C validator