Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnpbi | Structured version Visualization version GIF version |
Description: Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
Ref | Expression |
---|---|
sgnpbi | ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ∈ ℝ*) | |
2 | eqeq1 2828 | . . . . 5 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) = 1 ↔ 0 = 1)) | |
3 | 2 | imbi1d 344 | . . . 4 ⊢ ((sgn‘𝐴) = 0 → (((sgn‘𝐴) = 1 → 0 < 𝐴) ↔ (0 = 1 → 0 < 𝐴))) |
4 | eqeq1 2828 | . . . . 5 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) = 1 ↔ 1 = 1)) | |
5 | 4 | imbi1d 344 | . . . 4 ⊢ ((sgn‘𝐴) = 1 → (((sgn‘𝐴) = 1 → 0 < 𝐴) ↔ (1 = 1 → 0 < 𝐴))) |
6 | eqeq1 2828 | . . . . 5 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) = 1 ↔ -1 = 1)) | |
7 | 6 | imbi1d 344 | . . . 4 ⊢ ((sgn‘𝐴) = -1 → (((sgn‘𝐴) = 1 → 0 < 𝐴) ↔ (-1 = 1 → 0 < 𝐴))) |
8 | 0ne1 11711 | . . . . . . 7 ⊢ 0 ≠ 1 | |
9 | 8 | neii 3021 | . . . . . 6 ⊢ ¬ 0 = 1 |
10 | 9 | pm2.21i 119 | . . . . 5 ⊢ (0 = 1 → 0 < 𝐴) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (0 = 1 → 0 < 𝐴)) |
12 | simp2 1133 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴 ∧ 1 = 1) → 0 < 𝐴) | |
13 | 12 | 3expia 1117 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (1 = 1 → 0 < 𝐴)) |
14 | neg1rr 11755 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
15 | neg1lt0 11757 | . . . . . . . . 9 ⊢ -1 < 0 | |
16 | 0lt1 11165 | . . . . . . . . 9 ⊢ 0 < 1 | |
17 | 0re 10646 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
18 | 1re 10644 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
19 | 14, 17, 18 | lttri 10769 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1) |
20 | 15, 16, 19 | mp2an 690 | . . . . . . . 8 ⊢ -1 < 1 |
21 | 14, 20 | gtneii 10755 | . . . . . . 7 ⊢ 1 ≠ -1 |
22 | 21 | nesymi 3076 | . . . . . 6 ⊢ ¬ -1 = 1 |
23 | 22 | pm2.21i 119 | . . . . 5 ⊢ (-1 = 1 → 0 < 𝐴) |
24 | 23 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-1 = 1 → 0 < 𝐴)) |
25 | 1, 3, 5, 7, 11, 13, 24 | sgn3da 31803 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 → 0 < 𝐴)) |
26 | 25 | imp 409 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (sgn‘𝐴) = 1) → 0 < 𝐴) |
27 | sgnp 14452 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
28 | 26, 27 | impbida 799 | 1 ⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 0cc0 10540 1c1 10541 ℝ*cxr 10677 < clt 10678 -cneg 10874 sgncsgn 14448 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-sgn 14449 |
This theorem is referenced by: sgnmulsgp 31812 |
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