![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnnbi | Structured version Visualization version GIF version |
Description: Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
Ref | Expression |
---|---|
sgnnbi | ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ∈ ℝ*) | |
2 | eqeq1 2655 | . . . . 5 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = -1 ↔ 0 = -1)) | |
3 | 2 | imbi1d 330 | . . . 4 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (0 = -1 → 𝐴 < 0))) |
4 | eqeq1 2655 | . . . . 5 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = -1 ↔ 1 = -1)) | |
5 | 4 | imbi1d 330 | . . . 4 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (1 = -1 → 𝐴 < 0))) |
6 | eqeq1 2655 | . . . . 5 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = -1 ↔ -1 = -1)) | |
7 | 6 | imbi1d 330 | . . . 4 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (-1 = -1 → 𝐴 < 0))) |
8 | neg1ne0 11164 | . . . . . . 7 ⊢ -1 ≠ 0 | |
9 | 8 | nesymi 2880 | . . . . . 6 ⊢ ¬ 0 = -1 |
10 | 9 | pm2.21i 116 | . . . . 5 ⊢ (0 = -1 → 𝐴 < 0) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = -1 → 𝐴 < 0)) |
12 | neg1rr 11163 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
13 | neg1lt0 11165 | . . . . . . . . 9 ⊢ -1 < 0 | |
14 | 0lt1 10588 | . . . . . . . . 9 ⊢ 0 < 1 | |
15 | 0re 10078 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
16 | 1re 10077 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
17 | 12, 15, 16 | lttri 10201 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1) |
18 | 13, 14, 17 | mp2an 708 | . . . . . . . 8 ⊢ -1 < 1 |
19 | 12, 18 | gtneii 10187 | . . . . . . 7 ⊢ 1 ≠ -1 |
20 | 19 | neii 2825 | . . . . . 6 ⊢ ¬ 1 = -1 |
21 | 20 | pm2.21i 116 | . . . . 5 ⊢ (1 = -1 → 𝐴 < 0) |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = -1 → 𝐴 < 0)) |
23 | simp2 1082 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0 ∧ -1 = -1) → 𝐴 < 0) | |
24 | 23 | 3expia 1286 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = -1 → 𝐴 < 0)) |
25 | 1, 3, 5, 7, 11, 22, 24 | sgn3da 30731 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 → 𝐴 < 0)) |
26 | 25 | imp 444 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (sgn‘𝐴) = -1) → 𝐴 < 0) |
27 | sgnn 13878 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
28 | 26, 27 | impbida 895 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 0cc0 9974 1c1 9975 ℝ*cxr 10111 < clt 10112 -cneg 10305 sgncsgn 13870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-sgn 13871 |
This theorem is referenced by: sgnmulsgn 30739 |
Copyright terms: Public domain | W3C validator |