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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 12661 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | zltlesub.asb | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | |
4 | 3 | zred 12661 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
5 | zltlesub.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 4, 5 | readdcld 11238 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) ∈ ℝ) |
7 | peano2re 11382 | . . . 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 11237 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
12 | 5 | recnd 11237 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 11, 12 | npcand 11570 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
14 | 9, 13 | breqtrrd 5174 | . . 3 ⊢ (𝜑 → 𝑁 ≤ ((𝐴 − 𝐵) + 𝐵)) |
15 | 1red 11210 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | zltlesub.blt1 | . . . 4 ⊢ (𝜑 → 𝐵 < 1) | |
17 | 5, 15, 4, 16 | ltadd2dd 11368 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) < ((𝐴 − 𝐵) + 1)) |
18 | 2, 6, 8, 14, 17 | lelttrd 11367 | . 2 ⊢ (𝜑 → 𝑁 < ((𝐴 − 𝐵) + 1)) |
19 | zleltp1 12608 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) | |
20 | 1, 3, 19 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑁 ≤ (𝐴 − 𝐵) ↔ 𝑁 < ((𝐴 − 𝐵) + 1))) |
21 | 18, 20 | mpbird 257 | 1 ⊢ (𝜑 → 𝑁 ≤ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 class class class wbr 5146 (class class class)co 7403 ℝcr 11104 1c1 11106 + caddc 11108 < clt 11243 ≤ cle 11244 − cmin 11439 ℤcz 12553 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 |
This theorem is referenced by: fourierdlem65 44821 |
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