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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 12650 | . . 3 ⊢ -1 ∈ ℤ | |
2 | 1le1 11892 | . . 3 ⊢ 1 ≤ 1 | |
3 | fveq2 6901 | . . . . . 6 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1)) | |
4 | ax-1cn 11216 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
5 | 4 | absnegi 15405 | . . . . . . 7 ⊢ (abs‘-1) = (abs‘1) |
6 | abs1 15302 | . . . . . . 7 ⊢ (abs‘1) = 1 | |
7 | 5, 6 | eqtri 2754 | . . . . . 6 ⊢ (abs‘-1) = 1 |
8 | 3, 7 | eqtrdi 2782 | . . . . 5 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1) |
9 | 8 | breq1d 5163 | . . . 4 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
10 | lgslem2.z | . . . 4 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
11 | 9, 10 | elrab2 3684 | . . 3 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1)) |
12 | 1, 2, 11 | mpbir2an 709 | . 2 ⊢ -1 ∈ 𝑍 |
13 | 0z 12621 | . . 3 ⊢ 0 ∈ ℤ | |
14 | 0le1 11787 | . . 3 ⊢ 0 ≤ 1 | |
15 | fveq2 6901 | . . . . . 6 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0)) | |
16 | abs0 15290 | . . . . . 6 ⊢ (abs‘0) = 0 | |
17 | 15, 16 | eqtrdi 2782 | . . . . 5 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
18 | 17 | breq1d 5163 | . . . 4 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18, 10 | elrab2 3684 | . . 3 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1)) |
20 | 13, 14, 19 | mpbir2an 709 | . 2 ⊢ 0 ∈ 𝑍 |
21 | 1z 12644 | . . 3 ⊢ 1 ∈ ℤ | |
22 | fveq2 6901 | . . . . . 6 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1)) | |
23 | 22, 6 | eqtrdi 2782 | . . . . 5 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1) |
24 | 23 | breq1d 5163 | . . . 4 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1)) |
25 | 24, 10 | elrab2 3684 | . . 3 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1)) |
26 | 21, 2, 25 | mpbir2an 709 | . 2 ⊢ 1 ∈ 𝑍 |
27 | 12, 20, 26 | 3pm3.2i 1336 | 1 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {crab 3419 class class class wbr 5153 ‘cfv 6554 0cc0 11158 1c1 11159 ≤ cle 11299 -cneg 11495 ℤcz 12610 abscabs 15239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 |
This theorem is referenced by: lgslem4 27329 lgscllem 27333 |
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