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Mirrors > Home > ILE Home > Th. List > lgslem2 | 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 9274 | . . 3 ⊢ -1 ∈ ℤ | |
2 | 1le1 8519 | . . 3 ⊢ 1 ≤ 1 | |
3 | fveq2 5511 | . . . . . 6 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1)) | |
4 | ax-1cn 7895 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
5 | 4 | absnegi 11140 | . . . . . . 7 ⊢ (abs‘-1) = (abs‘1) |
6 | abs1 11065 | . . . . . . 7 ⊢ (abs‘1) = 1 | |
7 | 5, 6 | eqtri 2198 | . . . . . 6 ⊢ (abs‘-1) = 1 |
8 | 3, 7 | eqtrdi 2226 | . . . . 5 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1) |
9 | 8 | breq1d 4010 | . . . 4 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
10 | lgslem2.z | . . . 4 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
11 | 9, 10 | elrab2 2896 | . . 3 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1)) |
12 | 1, 2, 11 | mpbir2an 942 | . 2 ⊢ -1 ∈ 𝑍 |
13 | 0z 9253 | . . 3 ⊢ 0 ∈ ℤ | |
14 | 0le1 8428 | . . 3 ⊢ 0 ≤ 1 | |
15 | fveq2 5511 | . . . . . 6 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0)) | |
16 | abs0 11051 | . . . . . 6 ⊢ (abs‘0) = 0 | |
17 | 15, 16 | eqtrdi 2226 | . . . . 5 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
18 | 17 | breq1d 4010 | . . . 4 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18, 10 | elrab2 2896 | . . 3 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1)) |
20 | 13, 14, 19 | mpbir2an 942 | . 2 ⊢ 0 ∈ 𝑍 |
21 | 1z 9268 | . . 3 ⊢ 1 ∈ ℤ | |
22 | fveq2 5511 | . . . . . 6 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1)) | |
23 | 22, 6 | eqtrdi 2226 | . . . . 5 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1) |
24 | 23 | breq1d 4010 | . . . 4 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
25 | 24, 10 | elrab2 2896 | . . 3 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1)) |
26 | 21, 2, 25 | mpbir2an 942 | . 2 ⊢ 1 ∈ 𝑍 |
27 | 12, 20, 26 | 3pm3.2i 1175 | 1 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {crab 2459 class class class wbr 4000 ‘cfv 5212 0cc0 7802 1c1 7803 ≤ cle 7983 -cneg 8119 ℤcz 9242 abscabs 10990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-n0 9166 df-z 9243 df-uz 9518 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 |
This theorem is referenced by: lgslem4 14071 lgscllem 14075 |
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