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 13832

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 13803	. . . 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 12674	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		flle 13806	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
7	6	adantr 484	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴)
8	2, 3, 5, 7	leadd1dd 11798	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁))
9		1red 11179	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
10	2, 9	readdcld 11208	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ)
11		flltp1 13807	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
12	11	adantr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1))
13	3, 10, 5, 12	ltadd1dd 11795	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 1) + 𝑁))
14	2	recnd 11207	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℂ)
15		1cnd 11172	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℂ)
16	5	recnd 11207	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
17	14, 15, 16	add32d 11408	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (((⌊‘𝐴) + 1) + 𝑁) = (((⌊‘𝐴) + 𝑁) + 1))
18	13, 17	breqtrd 5125	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))
19	3, 5	readdcld 11208	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + 𝑁) ∈ ℝ)
20	3	flcld 13805	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ)
21	20, 4	zaddcld 12678	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + 𝑁) ∈ ℤ)
22		flbi 13823	. . 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 593	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁) ↔ (((⌊‘𝐴) + 𝑁) ≤ (𝐴 + 𝑁) ∧ (𝐴 + 𝑁) < (((⌊‘𝐴) + 𝑁) + 1))))
24	8, 18, 23	mpbir2and 723	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℝcr 11069 1c1 11071 + caddc 11073 < clt 11213 ≤ cle 11214 ℤcz 12565 ⌊cfl 13797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-n0 12479 df-z 12566 df-uz 12837 df-fl 13799

This theorem is referenced by: flzadd 13833 modcyc 13913 bitsmod 16453 fldivp1 16916 ppip1le 27202 dya2ub 34528 fourierdlem4 46649 fourierdlem47 46691 flsubz 49108 blennnt2 49175

W3C validator