| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > sgncl | Structured version Visualization version GIF version | ||
| Description: Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
| Ref | Expression |
|---|---|
| sgncl | ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
| 2 | 1 | fveq2d 6910 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = (sgn‘0)) |
| 3 | sgn0 15128 | . . . 4 ⊢ (sgn‘0) = 0 | |
| 4 | 2, 3 | eqtrdi 2793 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = 0) |
| 5 | c0ex 11255 | . . . 4 ⊢ 0 ∈ V | |
| 6 | 5 | tpid2 4770 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
| 7 | 4, 6 | eqeltrdi 2849 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 8 | sgnn 15133 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
| 9 | negex 11506 | . . . . . 6 ⊢ -1 ∈ V | |
| 10 | 9 | tpid1 4768 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
| 11 | 8, 10 | eqeltrdi 2849 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 12 | 11 | adantlr 715 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 13 | sgnp 15129 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
| 14 | 1ex 11257 | . . . . . 6 ⊢ 1 ∈ V | |
| 15 | 14 | tpid3 4773 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
| 16 | 13, 15 | eqeltrdi 2849 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 17 | 16 | adantlr 715 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 18 | 0xr 11308 | . . . 4 ⊢ 0 ∈ ℝ* | |
| 19 | xrlttri2 13184 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
| 20 | 19 | biimpa 476 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
| 21 | 18, 20 | mpanl2 701 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
| 22 | 12, 17, 21 | mpjaodan 961 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| 23 | 7, 22 | pm2.61dane 3029 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {ctp 4630 class class class wbr 5143 ‘cfv 6561 0cc0 11155 1c1 11156 ℝ*cxr 11294 < clt 11295 -cneg 11493 sgncsgn 15125 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-neg 11495 df-sgn 15126 |
| This theorem is referenced by: sgnclre 34542 sgnmulsgn 34552 sgnmulsgp 34553 signstcl 34580 signstf 34581 signstf0 34583 signstfvn 34584 signsvtn0 34585 signstfvneq0 34587 signsvfn 34597 |
| Copyright terms: Public domain | W3C validator |