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Mirrors > Home > MPE Home > Th. List > ceim1l | Structured version Visualization version GIF version |
Description: One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeff Hankins, 10-Jun-2007.) |
Ref | Expression |
---|---|
ceim1l | ⊢ (𝐴 ∈ ℝ → (-(⌊‘-𝐴) − 1) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl 11530 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | reflcl 13768 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ → (⌊‘-𝐴) ∈ ℝ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘-𝐴) ∈ ℝ) |
4 | 3 | recnd 11249 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘-𝐴) ∈ ℂ) |
5 | ax-1cn 11174 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | negdi 11524 | . . . 4 ⊢ (((⌊‘-𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → -((⌊‘-𝐴) + 1) = (-(⌊‘-𝐴) + -1)) | |
7 | 4, 5, 6 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ ℝ → -((⌊‘-𝐴) + 1) = (-(⌊‘-𝐴) + -1)) |
8 | 4 | negcld 11565 | . . . 4 ⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℂ) |
9 | negsub 11515 | . . . 4 ⊢ ((-(⌊‘-𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (-(⌊‘-𝐴) + -1) = (-(⌊‘-𝐴) − 1)) | |
10 | 8, 5, 9 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ ℝ → (-(⌊‘-𝐴) + -1) = (-(⌊‘-𝐴) − 1)) |
11 | 7, 10 | eqtr2d 2772 | . 2 ⊢ (𝐴 ∈ ℝ → (-(⌊‘-𝐴) − 1) = -((⌊‘-𝐴) + 1)) |
12 | peano2re 11394 | . . . 4 ⊢ ((⌊‘-𝐴) ∈ ℝ → ((⌊‘-𝐴) + 1) ∈ ℝ) | |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘-𝐴) + 1) ∈ ℝ) |
14 | flltp1 13772 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ → -𝐴 < ((⌊‘-𝐴) + 1)) | |
15 | 1, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 < ((⌊‘-𝐴) + 1)) |
16 | 15 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ ((⌊‘-𝐴) + 1) ∈ ℝ) → -𝐴 < ((⌊‘-𝐴) + 1)) |
17 | ltnegcon1 11722 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ ((⌊‘-𝐴) + 1) ∈ ℝ) → (-𝐴 < ((⌊‘-𝐴) + 1) ↔ -((⌊‘-𝐴) + 1) < 𝐴)) | |
18 | 16, 17 | mpbid 231 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ ((⌊‘-𝐴) + 1) ∈ ℝ) → -((⌊‘-𝐴) + 1) < 𝐴) |
19 | 13, 18 | mpdan 684 | . 2 ⊢ (𝐴 ∈ ℝ → -((⌊‘-𝐴) + 1) < 𝐴) |
20 | 11, 19 | eqbrtrd 5170 | 1 ⊢ (𝐴 ∈ ℝ → (-(⌊‘-𝐴) − 1) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 1c1 11117 + caddc 11119 < clt 11255 − cmin 11451 -cneg 11452 ⌊cfl 13762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fl 13764 |
This theorem is referenced by: ceilm1lt 13820 ceile 13821 ltflcei 36940 |
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