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Mirrors > Home > MPE Home > Th. List > Mathboxes > enege | Structured version Visualization version GIF version |
Description: The negative of an even number is even. (Contributed by AV, 20-Jun-2020.) |
Ref | Expression |
---|---|
enege | ⊢ (𝐴 ∈ Even → -𝐴 ∈ Even ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl 11747 | . . . 4 ⊢ (𝐴 ∈ ℤ → -𝐴 ∈ ℤ) | |
2 | 1 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → -𝐴 ∈ ℤ) |
3 | znegcl 11747 | . . . . 5 ⊢ ((𝐴 / 2) ∈ ℤ → -(𝐴 / 2) ∈ ℤ) | |
4 | 3 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → -(𝐴 / 2) ∈ ℤ) |
5 | zcn 11716 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
6 | 2cnd 11436 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℂ) | |
7 | 2ne0 11469 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 2 ≠ 0) |
9 | 5, 6, 8 | 3jca 1162 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0)) |
10 | 9 | adantr 474 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → (𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0)) |
11 | divneg 11051 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(𝐴 / 2) = (-𝐴 / 2)) | |
12 | 11 | eleq1d 2891 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (-(𝐴 / 2) ∈ ℤ ↔ (-𝐴 / 2) ∈ ℤ)) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → (-(𝐴 / 2) ∈ ℤ ↔ (-𝐴 / 2) ∈ ℤ)) |
14 | 4, 13 | mpbid 224 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → (-𝐴 / 2) ∈ ℤ) |
15 | 2, 14 | jca 507 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ) → (-𝐴 ∈ ℤ ∧ (-𝐴 / 2) ∈ ℤ)) |
16 | iseven 42385 | . 2 ⊢ (𝐴 ∈ Even ↔ (𝐴 ∈ ℤ ∧ (𝐴 / 2) ∈ ℤ)) | |
17 | iseven 42385 | . 2 ⊢ (-𝐴 ∈ Even ↔ (-𝐴 ∈ ℤ ∧ (-𝐴 / 2) ∈ ℤ)) | |
18 | 15, 16, 17 | 3imtr4i 284 | 1 ⊢ (𝐴 ∈ Even → -𝐴 ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 ∈ wcel 2164 ≠ wne 2999 (class class class)co 6910 ℂcc 10257 0cc0 10259 -cneg 10593 / cdiv 11016 2c2 11413 ℤcz 11711 Even ceven 42381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-z 11712 df-even 42383 |
This theorem is referenced by: omeoALTV 42441 emee 42459 |
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