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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnsgn | Structured version Visualization version GIF version |
Description: Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
Ref | Expression |
---|---|
sgnsgn | ⊢ (𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ∈ ℝ*) | |
2 | fveq2 6907 | . . 3 ⊢ ((sgn‘𝐴) = 0 → (sgn‘(sgn‘𝐴)) = (sgn‘0)) | |
3 | id 22 | . . 3 ⊢ ((sgn‘𝐴) = 0 → (sgn‘𝐴) = 0) | |
4 | 2, 3 | eqeq12d 2751 | . 2 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘(sgn‘𝐴)) = (sgn‘𝐴) ↔ (sgn‘0) = 0)) |
5 | fveq2 6907 | . . 3 ⊢ ((sgn‘𝐴) = 1 → (sgn‘(sgn‘𝐴)) = (sgn‘1)) | |
6 | id 22 | . . 3 ⊢ ((sgn‘𝐴) = 1 → (sgn‘𝐴) = 1) | |
7 | 5, 6 | eqeq12d 2751 | . 2 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘(sgn‘𝐴)) = (sgn‘𝐴) ↔ (sgn‘1) = 1)) |
8 | fveq2 6907 | . . 3 ⊢ ((sgn‘𝐴) = -1 → (sgn‘(sgn‘𝐴)) = (sgn‘-1)) | |
9 | id 22 | . . 3 ⊢ ((sgn‘𝐴) = -1 → (sgn‘𝐴) = -1) | |
10 | 8, 9 | eqeq12d 2751 | . 2 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘(sgn‘𝐴)) = (sgn‘𝐴) ↔ (sgn‘-1) = -1)) |
11 | sgn0 15125 | . . 3 ⊢ (sgn‘0) = 0 | |
12 | 11 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘0) = 0) |
13 | sgn1 15128 | . . 3 ⊢ (sgn‘1) = 1 | |
14 | 13 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘1) = 1) |
15 | neg1rr 12379 | . . . . 5 ⊢ -1 ∈ ℝ | |
16 | 15 | rexri 11317 | . . . 4 ⊢ -1 ∈ ℝ* |
17 | neg1lt0 12381 | . . . 4 ⊢ -1 < 0 | |
18 | sgnn 15130 | . . . 4 ⊢ ((-1 ∈ ℝ* ∧ -1 < 0) → (sgn‘-1) = -1) | |
19 | 16, 17, 18 | mp2an 692 | . . 3 ⊢ (sgn‘-1) = -1 |
20 | 19 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘-1) = -1) |
21 | 1, 4, 7, 10, 12, 14, 20 | sgn3da 34523 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 0cc0 11153 1c1 11154 ℝ*cxr 11292 < clt 11293 -cneg 11491 sgncsgn 15122 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-sgn 15123 |
This theorem is referenced by: signsvfn 34576 signsvfpn 34579 signsvfnn 34580 |
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