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 12247 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | zltlesub.asb | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | |
4 | 3 | zred 12247 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
5 | zltlesub.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 4, 5 | readdcld 10827 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) ∈ ℝ) |
7 | peano2re 10970 | . . . 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 10826 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
12 | 5 | recnd 10826 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 11, 12 | npcand 11158 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
14 | 9, 13 | breqtrrd 5067 | . . 3 ⊢ (𝜑 → 𝑁 ≤ ((𝐴 − 𝐵) + 𝐵)) |
15 | 1red 10799 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | zltlesub.blt1 | . . . 4 ⊢ (𝜑 → 𝐵 < 1) | |
17 | 5, 15, 4, 16 | ltadd2dd 10956 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) < ((𝐴 − 𝐵) + 1)) |
18 | 2, 6, 8, 14, 17 | lelttrd 10955 | . 2 ⊢ (𝜑 → 𝑁 < ((𝐴 − 𝐵) + 1)) |
19 | zleltp1 12193 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) | |
20 | 1, 3, 19 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) |
21 | 18, 20 | mpbird 260 | 1 ⊢ (𝜑 → 𝑁 ≤ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 1c1 10695 + caddc 10697 < clt 10832 ≤ cle 10833 − cmin 11027 ℤcz 12141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 |
This theorem is referenced by: fourierdlem65 43330 |
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