Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zltlesub | Structured version Visualization version GIF version |
Description: If an integer 𝑁 is less than or equal to a real, and we subtract a quantity less than 1, then 𝑁 is less than or equal to the result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
zltlesub.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
zltlesub.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
zltlesub.nlea | ⊢ (𝜑 → 𝑁 ≤ 𝐴) |
zltlesub.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
zltlesub.blt1 | ⊢ (𝜑 → 𝐵 < 1) |
zltlesub.asb | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
Ref | Expression |
---|---|
zltlesub | ⊢ (𝜑 → 𝑁 ≤ (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zltlesub.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
2 | 1 | zred 12081 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | zltlesub.asb | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | |
4 | 3 | zred 12081 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
5 | zltlesub.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 4, 5 | readdcld 10664 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) ∈ ℝ) |
7 | peano2re 10807 | . . . 4 ⊢ ((𝐴 − 𝐵) ∈ ℝ → ((𝐴 − 𝐵) + 1) ∈ ℝ) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 1) ∈ ℝ) |
9 | zltlesub.nlea | . . . 4 ⊢ (𝜑 → 𝑁 ≤ 𝐴) | |
10 | zltlesub.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
11 | 10 | recnd 10663 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
12 | 5 | recnd 10663 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 11, 12 | npcand 10995 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
14 | 9, 13 | breqtrrd 5086 | . . 3 ⊢ (𝜑 → 𝑁 ≤ ((𝐴 − 𝐵) + 𝐵)) |
15 | 1red 10636 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | zltlesub.blt1 | . . . 4 ⊢ (𝜑 → 𝐵 < 1) | |
17 | 5, 15, 4, 16 | ltadd2dd 10793 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) < ((𝐴 − 𝐵) + 1)) |
18 | 2, 6, 8, 14, 17 | lelttrd 10792 | . 2 ⊢ (𝜑 → 𝑁 < ((𝐴 − 𝐵) + 1)) |
19 | zleltp1 12027 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) | |
20 | 1, 3, 19 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) |
21 | 18, 20 | mpbird 259 | 1 ⊢ (𝜑 → 𝑁 ≤ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 ℤcz 11975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 |
This theorem is referenced by: fourierdlem65 42450 |
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