![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11703 | . . 3 ⊢ -1 ∈ ℤ | |
2 | 1le1 10947 | . . 3 ⊢ 1 ≤ 1 | |
3 | fveq2 6411 | . . . . . 6 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1)) | |
4 | ax-1cn 10282 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
5 | 4 | absnegi 14480 | . . . . . . 7 ⊢ (abs‘-1) = (abs‘1) |
6 | abs1 14378 | . . . . . . 7 ⊢ (abs‘1) = 1 | |
7 | 5, 6 | eqtri 2821 | . . . . . 6 ⊢ (abs‘-1) = 1 |
8 | 3, 7 | syl6eq 2849 | . . . . 5 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1) |
9 | 8 | breq1d 4853 | . . . 4 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
10 | lgslem2.z | . . . 4 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
11 | 9, 10 | elrab2 3560 | . . 3 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1)) |
12 | 1, 2, 11 | mpbir2an 703 | . 2 ⊢ -1 ∈ 𝑍 |
13 | 0z 11677 | . . 3 ⊢ 0 ∈ ℤ | |
14 | 0le1 10843 | . . 3 ⊢ 0 ≤ 1 | |
15 | fveq2 6411 | . . . . . 6 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0)) | |
16 | abs0 14366 | . . . . . 6 ⊢ (abs‘0) = 0 | |
17 | 15, 16 | syl6eq 2849 | . . . . 5 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
18 | 17 | breq1d 4853 | . . . 4 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18, 10 | elrab2 3560 | . . 3 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1)) |
20 | 13, 14, 19 | mpbir2an 703 | . 2 ⊢ 0 ∈ 𝑍 |
21 | 1z 11697 | . . 3 ⊢ 1 ∈ ℤ | |
22 | fveq2 6411 | . . . . . 6 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1)) | |
23 | 22, 6 | syl6eq 2849 | . . . . 5 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1) |
24 | 23 | breq1d 4853 | . . . 4 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
25 | 24, 10 | elrab2 3560 | . . 3 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1)) |
26 | 21, 2, 25 | mpbir2an 703 | . 2 ⊢ 1 ∈ 𝑍 |
27 | 12, 20, 26 | 3pm3.2i 1439 | 1 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {crab 3093 class class class wbr 4843 ‘cfv 6101 0cc0 10224 1c1 10225 ≤ cle 10364 -cneg 10557 ℤcz 11666 abscabs 14315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 |
This theorem is referenced by: lgslem4 25377 lgscllem 25381 |
Copyright terms: Public domain | W3C validator |