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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sgn0bi | Structured version Visualization version GIF version | ||
| Description: Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| sgn0bi | ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ∈ ℝ*) | |
| 2 | eqeq1 2740 | . . 3 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = 0 ↔ 0 = 0)) | |
| 3 | 2 | bibi1d 343 | . 2 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (0 = 0 ↔ 𝐴 = 0))) | 
| 4 | eqeq1 2740 | . . 3 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = 0 ↔ 1 = 0)) | |
| 5 | 4 | bibi1d 343 | . 2 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (1 = 0 ↔ 𝐴 = 0))) | 
| 6 | eqeq1 2740 | . . 3 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = 0 ↔ -1 = 0)) | |
| 7 | 6 | bibi1d 343 | . 2 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (-1 = 0 ↔ 𝐴 = 0))) | 
| 8 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
| 9 | 8 | eqcomd 2742 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 0 = 𝐴) | 
| 10 | 9 | eqeq1d 2738 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = 0 ↔ 𝐴 = 0)) | 
| 11 | ax-1ne0 11225 | . . . . 5 ⊢ 1 ≠ 0 | |
| 12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 1 ≠ 0) | 
| 13 | 12 | neneqd 2944 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 1 = 0) | 
| 14 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 0 < 𝐴) | |
| 15 | 14 | gt0ne0d 11828 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0) | 
| 16 | 15 | neneqd 2944 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 𝐴 = 0) | 
| 17 | 13, 16 | 2falsed 376 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = 0 ↔ 𝐴 = 0)) | 
| 18 | 1cnd 11257 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ∈ ℂ) | |
| 19 | 11 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ≠ 0) | 
| 20 | 18, 19 | negne0d 11619 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → -1 ≠ 0) | 
| 21 | 20 | neneqd 2944 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ -1 = 0) | 
| 22 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 < 0) | |
| 23 | 22 | lt0ne0d 11829 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 ≠ 0) | 
| 24 | 23 | neneqd 2944 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ 𝐴 = 0) | 
| 25 | 21, 24 | 2falsed 376 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = 0 ↔ 𝐴 = 0)) | 
| 26 | 1, 3, 5, 7, 10, 17, 25 | sgn3da 34545 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 class class class wbr 5142 ‘cfv 6560 0cc0 11156 1c1 11157 ℝ*cxr 11295 < clt 11296 -cneg 11494 sgncsgn 15126 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-sub 11495 df-neg 11496 df-sgn 15127 | 
| This theorem is referenced by: signsvtn0 34586 signstfvneq0 34588 | 
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