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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > flsubz | Structured version Visualization version GIF version |
Description: An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.) |
Ref | Expression |
---|---|
flsubz | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 − 𝑁)) = ((⌊‘𝐴) − 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11137 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | zcn 12500 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | negsub 11445 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 + -𝑁) = (𝐴 − 𝑁)) | |
4 | 1, 2, 3 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 + -𝑁) = (𝐴 − 𝑁)) |
5 | 4 | eqcomd 2742 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (𝐴 − 𝑁) = (𝐴 + -𝑁)) |
6 | 5 | fveq2d 6843 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 − 𝑁)) = (⌊‘(𝐴 + -𝑁))) |
7 | znegcl 12534 | . . 3 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
8 | fladdz 13722 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ -𝑁 ∈ ℤ) → (⌊‘(𝐴 + -𝑁)) = ((⌊‘𝐴) + -𝑁)) | |
9 | 7, 8 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + -𝑁)) = ((⌊‘𝐴) + -𝑁)) |
10 | reflcl 13693 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 11179 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℂ) |
12 | negsub 11445 | . . 3 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((⌊‘𝐴) + -𝑁) = ((⌊‘𝐴) − 𝑁)) | |
13 | 11, 2, 12 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → ((⌊‘𝐴) + -𝑁) = ((⌊‘𝐴) − 𝑁)) |
14 | 6, 9, 13 | 3eqtrd 2780 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 − 𝑁)) = ((⌊‘𝐴) − 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 ℂcc 11045 ℝcr 11046 + caddc 11050 − cmin 11381 -cneg 11382 ℤcz 12495 ⌊cfl 13687 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-fl 13689 |
This theorem is referenced by: blennngt2o2 46610 blennn0e2 46612 dignn0flhalflem2 46634 |
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