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Mirrors > Home > MPE Home > Th. List > Mathboxes > onego | Structured version Visualization version GIF version |
Description: The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.) |
Ref | Expression |
---|---|
onego | ⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl 11764 | . . . 4 ⊢ (𝐴 ∈ ℤ → -𝐴 ∈ ℤ) | |
2 | 1 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → -𝐴 ∈ ℤ) |
3 | znegcl 11764 | . . . . . 6 ⊢ (((𝐴 − 1) / 2) ∈ ℤ → -((𝐴 − 1) / 2) ∈ ℤ) | |
4 | 3 | adantl 475 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → -((𝐴 − 1) / 2) ∈ ℤ) |
5 | peano2zm 11772 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℤ) | |
6 | 5 | zcnd 11835 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℂ) |
7 | 6 | adantr 474 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (𝐴 − 1) ∈ ℂ) |
8 | 2cnd 11453 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → 2 ∈ ℂ) | |
9 | 2ne0 11486 | . . . . . . 7 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → 2 ≠ 0) |
11 | divneg 11067 | . . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -((𝐴 − 1) / 2) = (-(𝐴 − 1) / 2)) | |
12 | 11 | eleq1d 2844 | . . . . . 6 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (-((𝐴 − 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
13 | 7, 8, 10, 12 | syl3anc 1439 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-((𝐴 − 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
14 | 4, 13 | mpbid 224 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-(𝐴 − 1) / 2) ∈ ℤ) |
15 | zcn 11733 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
16 | 1cnd 10371 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℂ) | |
17 | negsubdi 10679 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (-𝐴 + 1)) | |
18 | 17 | eqcomd 2784 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (-𝐴 + 1) = -(𝐴 − 1)) |
19 | 15, 16, 18 | syl2anc 579 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (-𝐴 + 1) = -(𝐴 − 1)) |
20 | 19 | oveq1d 6937 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → ((-𝐴 + 1) / 2) = (-(𝐴 − 1) / 2)) |
21 | 20 | eleq1d 2844 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (((-𝐴 + 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
22 | 21 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (((-𝐴 + 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
23 | 14, 22 | mpbird 249 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → ((-𝐴 + 1) / 2) ∈ ℤ) |
24 | 2, 23 | jca 507 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-𝐴 ∈ ℤ ∧ ((-𝐴 + 1) / 2) ∈ ℤ)) |
25 | isodd2 42577 | . 2 ⊢ (𝐴 ∈ Odd ↔ (𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ)) | |
26 | isodd 42571 | . 2 ⊢ (-𝐴 ∈ Odd ↔ (-𝐴 ∈ ℤ ∧ ((-𝐴 + 1) / 2) ∈ ℤ)) | |
27 | 24, 25, 26 | 3imtr4i 284 | 1 ⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 + caddc 10275 − cmin 10606 -cneg 10607 / cdiv 11032 2c2 11430 ℤcz 11728 Odd codd 42567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-odd 42569 |
This theorem is referenced by: omoeALTV 42625 emoo 42642 |
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