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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 11605 | . . 3 ⊢ -1 ∈ ℤ | |
2 | 1le1 10847 | . . 3 ⊢ 1 ≤ 1 | |
3 | fveq2 6352 | . . . . . 6 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1)) | |
4 | ax-1cn 10186 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
5 | 4 | absnegi 14338 | . . . . . . 7 ⊢ (abs‘-1) = (abs‘1) |
6 | abs1 14236 | . . . . . . 7 ⊢ (abs‘1) = 1 | |
7 | 5, 6 | eqtri 2782 | . . . . . 6 ⊢ (abs‘-1) = 1 |
8 | 3, 7 | syl6eq 2810 | . . . . 5 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1) |
9 | 8 | breq1d 4814 | . . . 4 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
10 | lgslem2.z | . . . 4 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
11 | 9, 10 | elrab2 3507 | . . 3 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1)) |
12 | 1, 2, 11 | mpbir2an 993 | . 2 ⊢ -1 ∈ 𝑍 |
13 | 0z 11580 | . . 3 ⊢ 0 ∈ ℤ | |
14 | 0le1 10743 | . . 3 ⊢ 0 ≤ 1 | |
15 | fveq2 6352 | . . . . . 6 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0)) | |
16 | abs0 14224 | . . . . . 6 ⊢ (abs‘0) = 0 | |
17 | 15, 16 | syl6eq 2810 | . . . . 5 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
18 | 17 | breq1d 4814 | . . . 4 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18, 10 | elrab2 3507 | . . 3 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1)) |
20 | 13, 14, 19 | mpbir2an 993 | . 2 ⊢ 0 ∈ 𝑍 |
21 | 1z 11599 | . . 3 ⊢ 1 ∈ ℤ | |
22 | fveq2 6352 | . . . . . 6 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1)) | |
23 | 22, 6 | syl6eq 2810 | . . . . 5 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1) |
24 | 23 | breq1d 4814 | . . . 4 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
25 | 24, 10 | elrab2 3507 | . . 3 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1)) |
26 | 21, 2, 25 | mpbir2an 993 | . 2 ⊢ 1 ∈ 𝑍 |
27 | 12, 20, 26 | 3pm3.2i 1424 | 1 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 {crab 3054 class class class wbr 4804 ‘cfv 6049 0cc0 10128 1c1 10129 ≤ cle 10267 -cneg 10459 ℤcz 11569 abscabs 14173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 |
This theorem is referenced by: lgslem4 25224 lgscllem 25228 |
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