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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 9884 | . . . 4 ⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) | |
2 | reex 7107 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2 | mptex 5408 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) ∈ V |
4 | 1, 3 | eqeltri 2151 | . . 3 ⊢ √ ∈ V |
5 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
6 | cjcl 9735 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
7 | 5, 6 | mulcld 7139 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℂ) |
8 | fvexg 5214 | . . 3 ⊢ ((√ ∈ V ∧ (𝐴 · (∗‘𝐴)) ∈ ℂ) → (√‘(𝐴 · (∗‘𝐴))) ∈ V) | |
9 | 4, 7, 8 | sylancr 405 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ V) |
10 | fveq2 5198 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
11 | oveq12 5541 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) | |
12 | 10, 11 | mpdan 412 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) |
13 | 12 | fveq2d 5202 | . . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴)))) |
14 | df-abs 9885 | . . 3 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
15 | 13, 14 | fvmptg 5269 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (√‘(𝐴 · (∗‘𝐴))) ∈ V) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
16 | 9, 15 | mpdan 412 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 class class class wbr 3785 ↦ cmpt 3839 ‘cfv 4922 ℩crio 5487 (class class class)co 5532 ℂcc 6979 ℝcr 6980 0cc0 6981 · cmul 6986 ≤ cle 7154 2c2 8089 ↑cexp 9475 ∗ccj 9726 √csqrt 9882 abscabs 9883 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-cj 9729 df-rsqrt 9884 df-abs 9885 |
This theorem is referenced by: absneg 9936 abscl 9937 abscj 9938 absvalsq 9939 absval2 9943 abs0 9944 absi 9945 absge0 9946 absrpclap 9947 absmul 9955 absid 9957 absre 9963 absf 9996 |
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