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Mathbox for Thierry Arnoux |
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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 2830 | . . 3 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = 0 ↔ 0 = 0)) | |
3 | 2 | bibi1d 335 | . 2 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (0 = 0 ↔ 𝐴 = 0))) |
4 | eqeq1 2830 | . . 3 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = 0 ↔ 1 = 0)) | |
5 | 4 | bibi1d 335 | . 2 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (1 = 0 ↔ 𝐴 = 0))) |
6 | eqeq1 2830 | . . 3 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = 0 ↔ -1 = 0)) | |
7 | 6 | bibi1d 335 | . 2 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = 0 ↔ 𝐴 = 0) ↔ (-1 = 0 ↔ 𝐴 = 0))) |
8 | simpr 479 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
9 | 8 | eqcomd 2832 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 0 = 𝐴) |
10 | 9 | eqeq1d 2828 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = 0 ↔ 𝐴 = 0)) |
11 | ax-1ne0 10322 | . . . . 5 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 1 ≠ 0) |
13 | 12 | neneqd 3005 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 1 = 0) |
14 | simpr 479 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 0 < 𝐴) | |
15 | 14 | gt0ne0d 10917 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
16 | 15 | neneqd 3005 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 𝐴 = 0) |
17 | 13, 16 | 2falsed 368 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = 0 ↔ 𝐴 = 0)) |
18 | 1cnd 10352 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ∈ ℂ) | |
19 | 11 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 1 ≠ 0) |
20 | 18, 19 | negne0d 10712 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → -1 ≠ 0) |
21 | 20 | neneqd 3005 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ -1 = 0) |
22 | simpr 479 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 < 0) | |
23 | 22 | lt0ne0d 10918 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 ≠ 0) |
24 | 23 | neneqd 3005 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ 𝐴 = 0) |
25 | 21, 24 | 2falsed 368 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = 0 ↔ 𝐴 = 0)) |
26 | 1, 3, 5, 7, 10, 17, 25 | sgn3da 31150 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 class class class wbr 4874 ‘cfv 6124 0cc0 10253 1c1 10254 ℝ*cxr 10391 < clt 10392 -cneg 10587 sgncsgn 14204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-sub 10588 df-neg 10589 df-sgn 14205 |
This theorem is referenced by: signsvtn0 31195 signsvtn0OLD 31196 signstfvneq0 31198 |
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