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| Mirrors > Home > MPE Home > Th. List > eldmgm | Structured version Visualization version GIF version | ||
| Description: Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.) |
| Ref | Expression |
|---|---|
| eldmgm | ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldif 3961 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ))) | |
| 2 | eldif 3961 | . . . . 5 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ)) | |
| 3 | elznn 12629 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℝ ∧ (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0))) | |
| 4 | 3 | simprbi 496 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0)) |
| 5 | 4 | orcanai 1005 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) → -𝐴 ∈ ℕ0) |
| 6 | negneg 11559 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
| 7 | 6 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 = 𝐴) |
| 8 | nn0negz 12655 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ0 → --𝐴 ∈ ℤ) | |
| 9 | 8 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 ∈ ℤ) |
| 10 | 7, 9 | eqeltrrd 2842 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℤ) |
| 11 | 10 | ex 412 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)) |
| 12 | nngt0 12297 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
| 13 | nnre 12273 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
| 14 | 13 | lt0neg2d 11833 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (0 < 𝐴 ↔ -𝐴 < 0)) |
| 15 | 12, 14 | mpbid 232 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -𝐴 < 0) |
| 16 | 13 | renegcld 11690 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → -𝐴 ∈ ℝ) |
| 17 | 0re 11263 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
| 18 | ltnle 11340 | . . . . . . . . . 10 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) | |
| 19 | 16, 17, 18 | sylancl 586 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) |
| 20 | 15, 19 | mpbid 232 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 0 ≤ -𝐴) |
| 21 | nn0ge0 12551 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℕ0 → 0 ≤ -𝐴) | |
| 22 | 20, 21 | nsyl3 138 | . . . . . . 7 ⊢ (-𝐴 ∈ ℕ0 → ¬ 𝐴 ∈ ℕ) |
| 23 | 11, 22 | jca2 513 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ))) |
| 24 | 5, 23 | impbid2 226 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) ↔ -𝐴 ∈ ℕ0)) |
| 25 | 2, 24 | bitrid 283 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ (ℤ ∖ ℕ) ↔ -𝐴 ∈ ℕ0)) |
| 26 | 25 | notbid 318 | . . 3 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ (ℤ ∖ ℕ) ↔ ¬ -𝐴 ∈ ℕ0)) |
| 27 | 26 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
| 28 | 1, 27 | bitri 275 | 1 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 class class class wbr 5143 ℂcc 11153 ℝcr 11154 0cc0 11155 < clt 11295 ≤ cle 11296 -cneg 11493 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 |
| This theorem is referenced by: dmgmaddn0 27066 dmlogdmgm 27067 dmgmaddnn0 27070 lgamgulmlem1 27072 lgamucov 27081 |
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