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Mirrors > Home > MPE Home > Th. List > lgslem2 | Structured version Visualization version GIF version |
Description: The set 𝑍 of all integers with absolute value at most 1 contains {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
lgslem2.z | ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} |
Ref | Expression |
---|---|
lgslem2 | ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1z 11867 | . . 3 ⊢ -1 ∈ ℤ | |
2 | 1le1 11116 | . . 3 ⊢ 1 ≤ 1 | |
3 | fveq2 6538 | . . . . . 6 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1)) | |
4 | ax-1cn 10441 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
5 | 4 | absnegi 14594 | . . . . . . 7 ⊢ (abs‘-1) = (abs‘1) |
6 | abs1 14491 | . . . . . . 7 ⊢ (abs‘1) = 1 | |
7 | 5, 6 | eqtri 2819 | . . . . . 6 ⊢ (abs‘-1) = 1 |
8 | 3, 7 | syl6eq 2847 | . . . . 5 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1) |
9 | 8 | breq1d 4972 | . . . 4 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
10 | lgslem2.z | . . . 4 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
11 | 9, 10 | elrab2 3621 | . . 3 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1)) |
12 | 1, 2, 11 | mpbir2an 707 | . 2 ⊢ -1 ∈ 𝑍 |
13 | 0z 11840 | . . 3 ⊢ 0 ∈ ℤ | |
14 | 0le1 11011 | . . 3 ⊢ 0 ≤ 1 | |
15 | fveq2 6538 | . . . . . 6 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0)) | |
16 | abs0 14479 | . . . . . 6 ⊢ (abs‘0) = 0 | |
17 | 15, 16 | syl6eq 2847 | . . . . 5 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
18 | 17 | breq1d 4972 | . . . 4 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18, 10 | elrab2 3621 | . . 3 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1)) |
20 | 13, 14, 19 | mpbir2an 707 | . 2 ⊢ 0 ∈ 𝑍 |
21 | 1z 11861 | . . 3 ⊢ 1 ∈ ℤ | |
22 | fveq2 6538 | . . . . . 6 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1)) | |
23 | 22, 6 | syl6eq 2847 | . . . . 5 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1) |
24 | 23 | breq1d 4972 | . . . 4 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
25 | 24, 10 | elrab2 3621 | . . 3 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1)) |
26 | 21, 2, 25 | mpbir2an 707 | . 2 ⊢ 1 ∈ 𝑍 |
27 | 12, 20, 26 | 3pm3.2i 1332 | 1 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 {crab 3109 class class class wbr 4962 ‘cfv 6225 0cc0 10383 1c1 10384 ≤ cle 10522 -cneg 10718 ℤcz 11829 abscabs 14427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 |
This theorem is referenced by: lgslem4 25558 lgscllem 25562 |
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