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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 10562 | . . . 4 ⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) | |
2 | reex 7573 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2 | mptex 5562 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) ∈ V |
4 | 1, 3 | eqeltri 2167 | . . 3 ⊢ √ ∈ V |
5 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
6 | cjcl 10413 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
7 | 5, 6 | mulcld 7605 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℂ) |
8 | fvexg 5359 | . . 3 ⊢ ((√ ∈ V ∧ (𝐴 · (∗‘𝐴)) ∈ ℂ) → (√‘(𝐴 · (∗‘𝐴))) ∈ V) | |
9 | 4, 7, 8 | sylancr 406 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ V) |
10 | fveq2 5340 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
11 | oveq12 5699 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) | |
12 | 10, 11 | mpdan 413 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) |
13 | 12 | fveq2d 5344 | . . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴)))) |
14 | df-abs 10563 | . . 3 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
15 | 13, 14 | fvmptg 5415 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (√‘(𝐴 · (∗‘𝐴))) ∈ V) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
16 | 9, 15 | mpdan 413 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 class class class wbr 3867 ↦ cmpt 3921 ‘cfv 5049 ℩crio 5645 (class class class)co 5690 ℂcc 7445 ℝcr 7446 0cc0 7447 · cmul 7452 ≤ cle 7620 2c2 8571 ↑cexp 10085 ∗ccj 10404 √csqrt 10560 abscabs 10561 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-ltxr 7624 df-sub 7752 df-neg 7753 df-reap 8149 df-cj 10407 df-rsqrt 10562 df-abs 10563 |
This theorem is referenced by: absneg 10614 abscl 10615 abscj 10616 absvalsq 10617 absval2 10621 abs0 10622 absi 10623 absge0 10624 absrpclap 10625 absmul 10633 absid 10635 absre 10641 absf 10674 |
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