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Mirrors > Home > MPE Home > Th. List > rpneg | Structured version Visualization version GIF version |
Description: Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
rpneg | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 ∈ ℝ+ ↔ ¬ -𝐴 ∈ ℝ+)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11200 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
2 | ltle 11286 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
3 | 1, 2 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → 0 ≤ 𝐴)) |
4 | 3 | imp 407 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 𝐴) |
5 | 4 | olcd 872 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (¬ -𝐴 ∈ ℝ ∨ 0 ≤ 𝐴)) |
6 | renegcl 11507 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
7 | 6 | pm2.24d 151 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (¬ -𝐴 ∈ ℝ → 0 < 𝐴)) |
8 | 7 | adantr 481 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (¬ -𝐴 ∈ ℝ → 0 < 𝐴)) |
9 | ltlen 11299 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 ≠ 0))) | |
10 | 1, 9 | mpan 688 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 ≠ 0))) |
11 | 10 | biimprd 247 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ((0 ≤ 𝐴 ∧ 𝐴 ≠ 0) → 0 < 𝐴)) |
12 | 11 | expcomd 417 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 ≠ 0 → (0 ≤ 𝐴 → 0 < 𝐴))) |
13 | 12 | imp 407 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 ≤ 𝐴 → 0 < 𝐴)) |
14 | 8, 13 | jaod 857 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((¬ -𝐴 ∈ ℝ ∨ 0 ≤ 𝐴) → 0 < 𝐴)) |
15 | simpl 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℝ) | |
16 | 14, 15 | jctild 526 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((¬ -𝐴 ∈ ℝ ∨ 0 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))) |
17 | 5, 16 | impbid2 225 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ↔ (¬ -𝐴 ∈ ℝ ∨ 0 ≤ 𝐴))) |
18 | lenlt 11276 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) | |
19 | 1, 18 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) |
20 | lt0neg1 11704 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
21 | 20 | notbid 317 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 0 ↔ ¬ 0 < -𝐴)) |
22 | 19, 21 | bitrd 278 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ ¬ 0 < -𝐴)) |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 ≤ 𝐴 ↔ ¬ 0 < -𝐴)) |
24 | 23 | orbi2d 914 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((¬ -𝐴 ∈ ℝ ∨ 0 ≤ 𝐴) ↔ (¬ -𝐴 ∈ ℝ ∨ ¬ 0 < -𝐴))) |
25 | 17, 24 | bitrd 278 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ↔ (¬ -𝐴 ∈ ℝ ∨ ¬ 0 < -𝐴))) |
26 | ianor 980 | . . 3 ⊢ (¬ (-𝐴 ∈ ℝ ∧ 0 < -𝐴) ↔ (¬ -𝐴 ∈ ℝ ∨ ¬ 0 < -𝐴)) | |
27 | 25, 26 | bitr4di 288 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ↔ ¬ (-𝐴 ∈ ℝ ∧ 0 < -𝐴))) |
28 | elrp 12960 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
29 | elrp 12960 | . . 3 ⊢ (-𝐴 ∈ ℝ+ ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) | |
30 | 29 | notbii 319 | . 2 ⊢ (¬ -𝐴 ∈ ℝ+ ↔ ¬ (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) |
31 | 27, 28, 30 | 3bitr4g 313 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 ∈ ℝ+ ↔ ¬ -𝐴 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5142 ℝcr 11093 0cc0 11094 < clt 11232 ≤ cle 11233 -cneg 11429 ℝ+crp 12958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-rp 12959 |
This theorem is referenced by: cnpart 15171 angpined 26264 signsply0 33457 |
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