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Mirrors > Home > ILE Home > Th. List > qabsor | GIF version |
Description: The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.) |
Ref | Expression |
---|---|
qabsor | ⊢ (𝐴 ∈ ℚ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qre 9444 | . 2 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
2 | 0z 9089 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | zq 9445 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℚ) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 0 ∈ ℚ |
5 | qletric 10052 | . . 3 ⊢ ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0)) | |
6 | 4, 5 | mpan 421 | . 2 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0)) |
7 | absid 10875 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) | |
8 | 7 | ex 114 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)) |
9 | absnid 10877 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (abs‘𝐴) = -𝐴) | |
10 | 9 | ex 114 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴)) |
11 | 8, 10 | orim12d 776 | . 2 ⊢ (𝐴 ∈ ℝ → ((0 ≤ 𝐴 ∨ 𝐴 ≤ 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))) |
12 | 1, 6, 11 | sylc 62 | 1 ⊢ (𝐴 ∈ ℚ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 ℝcr 7643 0cc0 7644 ≤ cle 7825 -cneg 7958 ℤcz 9078 ℚcq 9438 abscabs 10801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 |
This theorem is referenced by: qabsord 10880 gcdabs 11712 lcmabs 11793 |
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