Mathbox for Thierry Arnoux |
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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 2802 | . . . . 5 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = -1 ↔ 0 = -1)) | |
3 | 2 | imbi1d 345 | . . . 4 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (0 = -1 → 𝐴 < 0))) |
4 | eqeq1 2802 | . . . . 5 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = -1 ↔ 1 = -1)) | |
5 | 4 | imbi1d 345 | . . . 4 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (1 = -1 → 𝐴 < 0))) |
6 | eqeq1 2802 | . . . . 5 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = -1 ↔ -1 = -1)) | |
7 | 6 | imbi1d 345 | . . . 4 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = -1 → 𝐴 < 0) ↔ (-1 = -1 → 𝐴 < 0))) |
8 | neg1ne0 11741 | . . . . . . 7 ⊢ -1 ≠ 0 | |
9 | 8 | nesymi 3044 | . . . . . 6 ⊢ ¬ 0 = -1 |
10 | 9 | pm2.21i 119 | . . . . 5 ⊢ (0 = -1 → 𝐴 < 0) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = -1 → 𝐴 < 0)) |
12 | neg1rr 11740 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
13 | neg1lt0 11742 | . . . . . . . . 9 ⊢ -1 < 0 | |
14 | 0lt1 11151 | . . . . . . . . 9 ⊢ 0 < 1 | |
15 | 0re 10632 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
16 | 1re 10630 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
17 | 12, 15, 16 | lttri 10755 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1) |
18 | 13, 14, 17 | mp2an 691 | . . . . . . . 8 ⊢ -1 < 1 |
19 | 12, 18 | gtneii 10741 | . . . . . . 7 ⊢ 1 ≠ -1 |
20 | 19 | neii 2989 | . . . . . 6 ⊢ ¬ 1 = -1 |
21 | 20 | pm2.21i 119 | . . . . 5 ⊢ (1 = -1 → 𝐴 < 0) |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = -1 → 𝐴 < 0)) |
23 | simp2 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0 ∧ -1 = -1) → 𝐴 < 0) | |
24 | 23 | 3expia 1118 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = -1 → 𝐴 < 0)) |
25 | 1, 3, 5, 7, 11, 22, 24 | sgn3da 31909 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 → 𝐴 < 0)) |
26 | 25 | imp 410 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (sgn‘𝐴) = -1) → 𝐴 < 0) |
27 | sgnn 14445 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
28 | 26, 27 | impbida 800 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 ℝ*cxr 10663 < clt 10664 -cneg 10860 sgncsgn 14437 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-sgn 14438 |
This theorem is referenced by: sgnmulsgn 31917 |
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