fladdz - Metamath Proof Explorer

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Theorem fladdz 13545

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 13516	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
2	1	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ)
3		simpl 483	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ)
4		simpr 485	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
5	4	zred 12426	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		flle 13519	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
7	6	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴)
8	2, 3, 5, 7	leadd1dd 11589	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁))
9		1red 10976	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
10	2, 9	readdcld 11004	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ)
11		flltp1 13520	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
12	11	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1))
13	3, 10, 5, 12	ltadd1dd 11586	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 1) + 𝑁))
14	2	recnd 11003	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℂ)
15		1cnd 10970	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℂ)
16	5	recnd 11003	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
17	14, 15, 16	add32d 11202	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (((⌊‘𝐴) + 1) + 𝑁) = (((⌊‘𝐴) + 𝑁) + 1))
18	13, 17	breqtrd 5100	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))
19	3, 5	readdcld 11004	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) ∈ ℝ)
20	3	flcld 13518	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ)
21	20, 4	zaddcld 12430	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ∈ ℤ)
22		flbi 13536	. . 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 584	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁) ↔ (((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁) ∧ (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))))
24	8, 18, 23	mpbir2and 710	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 ℤcz 12319 ⌊cfl 13510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fl 13512

This theorem is referenced by: flzadd 13546 modcyc 13626 bitsmod 16143 fldivp1 16598 ppip1le 26310 dya2ub 32237 fourierdlem4 43652 fourierdlem47 43694 flsubz 45863 blennnt2 45935

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