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Mirrors > Home > ILE Home > Th. List > absval | GIF version |
Description: The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
absval | ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rsqrt 10802 | . . . 4 ⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) | |
2 | reex 7778 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2 | mptex 5654 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) ∈ V |
4 | 1, 3 | eqeltri 2213 | . . 3 ⊢ √ ∈ V |
5 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
6 | cjcl 10652 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
7 | 5, 6 | mulcld 7810 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℂ) |
8 | fvexg 5448 | . . 3 ⊢ ((√ ∈ V ∧ (𝐴 · (∗‘𝐴)) ∈ ℂ) → (√‘(𝐴 · (∗‘𝐴))) ∈ V) | |
9 | 4, 7, 8 | sylancr 411 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ V) |
10 | fveq2 5429 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
11 | oveq12 5791 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) | |
12 | 10, 11 | mpdan 418 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) |
13 | 12 | fveq2d 5433 | . . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴)))) |
14 | df-abs 10803 | . . 3 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
15 | 13, 14 | fvmptg 5505 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (√‘(𝐴 · (∗‘𝐴))) ∈ V) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
16 | 9, 15 | mpdan 418 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 class class class wbr 3937 ↦ cmpt 3997 ‘cfv 5131 ℩crio 5737 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 · cmul 7649 ≤ cle 7825 2c2 8795 ↑cexp 10323 ∗ccj 10643 √csqrt 10800 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-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 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-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 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-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 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-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-cj 10646 df-rsqrt 10802 df-abs 10803 |
This theorem is referenced by: absneg 10854 abscl 10855 abscj 10856 absvalsq 10857 absval2 10861 abs0 10862 absi 10863 absge0 10864 absrpclap 10865 absmul 10873 absid 10875 absre 10881 absf 10914 |
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