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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 12020 | . . . 4 ⊢ (𝐴 ∈ ℤ → -𝐴 ∈ ℤ) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → -𝐴 ∈ ℤ) |
3 | znegcl 12020 | . . . . . 6 ⊢ (((𝐴 − 1) / 2) ∈ ℤ → -((𝐴 − 1) / 2) ∈ ℤ) | |
4 | 3 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → -((𝐴 − 1) / 2) ∈ ℤ) |
5 | peano2zm 12028 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℤ) | |
6 | 5 | zcnd 12091 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℂ) |
7 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (𝐴 − 1) ∈ ℂ) |
8 | 2cnd 11718 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → 2 ∈ ℂ) | |
9 | 2ne0 11744 | . . . . . . 7 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → 2 ≠ 0) |
11 | divneg 11335 | . . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -((𝐴 − 1) / 2) = (-(𝐴 − 1) / 2)) | |
12 | 11 | eleq1d 2900 | . . . . . 6 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (-((𝐴 − 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
13 | 7, 8, 10, 12 | syl3anc 1367 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-((𝐴 − 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
14 | 4, 13 | mpbid 234 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-(𝐴 − 1) / 2) ∈ ℤ) |
15 | zcn 11989 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
16 | 1cnd 10639 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℂ) | |
17 | negsubdi 10945 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (-𝐴 + 1)) | |
18 | 17 | eqcomd 2830 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (-𝐴 + 1) = -(𝐴 − 1)) |
19 | 15, 16, 18 | syl2anc 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (-𝐴 + 1) = -(𝐴 − 1)) |
20 | 19 | oveq1d 7174 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → ((-𝐴 + 1) / 2) = (-(𝐴 − 1) / 2)) |
21 | 20 | eleq1d 2900 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (((-𝐴 + 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (((-𝐴 + 1) / 2) ∈ ℤ ↔ (-(𝐴 − 1) / 2) ∈ ℤ)) |
23 | 14, 22 | mpbird 259 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → ((-𝐴 + 1) / 2) ∈ ℤ) |
24 | 2, 23 | jca 514 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ) → (-𝐴 ∈ ℤ ∧ ((-𝐴 + 1) / 2) ∈ ℤ)) |
25 | isodd2 43807 | . 2 ⊢ (𝐴 ∈ Odd ↔ (𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℤ)) | |
26 | isodd 43801 | . 2 ⊢ (-𝐴 ∈ Odd ↔ (-𝐴 ∈ ℤ ∧ ((-𝐴 + 1) / 2) ∈ ℤ)) | |
27 | 24, 25, 26 | 3imtr4i 294 | 1 ⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 − cmin 10873 -cneg 10874 / cdiv 11300 2c2 11695 ℤcz 11984 Odd codd 43797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-odd 43799 |
This theorem is referenced by: omoeALTV 43857 emoo 43876 |
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