Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2825 | . . 3 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = 0 ↔ 0 = 0)) | |
3 | 2 | bibi1d 346 | . 2 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (0 = 0 ↔ 𝐴 = 0))) |
4 | eqeq1 2825 | . . 3 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = 0 ↔ 1 = 0)) | |
5 | 4 | bibi1d 346 | . 2 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (1 = 0 ↔ 𝐴 = 0))) |
6 | eqeq1 2825 | . . 3 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = 0 ↔ -1 = 0)) | |
7 | 6 | bibi1d 346 | . 2 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (-1 = 0 ↔ 𝐴 = 0))) |
8 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
9 | 8 | eqcomd 2827 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 0 = 𝐴) |
10 | 9 | eqeq1d 2823 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = 0 ↔ 𝐴 = 0)) |
11 | ax-1ne0 10600 | . . . . 5 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 1 ≠ 0) |
13 | 12 | neneqd 3021 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 1 = 0) |
14 | simpr 487 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 0 < 𝐴) | |
15 | 14 | gt0ne0d 11198 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
16 | 15 | neneqd 3021 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 𝐴 = 0) |
17 | 13, 16 | 2falsed 379 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = 0 ↔ 𝐴 = 0)) |
18 | 1cnd 10630 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ∈ ℂ) | |
19 | 11 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ≠ 0) |
20 | 18, 19 | negne0d 10989 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → -1 ≠ 0) |
21 | 20 | neneqd 3021 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ -1 = 0) |
22 | simpr 487 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 < 0) | |
23 | 22 | lt0ne0d 11199 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 ≠ 0) |
24 | 23 | neneqd 3021 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ 𝐴 = 0) |
25 | 21, 24 | 2falsed 379 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = 0 ↔ 𝐴 = 0)) |
26 | 1, 3, 5, 7, 10, 17, 25 | sgn3da 31794 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 0cc0 10531 1c1 10532 ℝ*cxr 10668 < clt 10669 -cneg 10865 sgncsgn 14439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-sub 10866 df-neg 10867 df-sgn 14440 |
This theorem is referenced by: signsvtn0 31835 signstfvneq0 31837 |
Copyright terms: Public domain | W3C validator |