![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 11042 | . . . 4 ⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) | |
2 | reex 7976 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2 | mptex 5763 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) ∈ V |
4 | 1, 3 | eqeltri 2262 | . . 3 ⊢ √ ∈ V |
5 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
6 | cjcl 10892 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
7 | 5, 6 | mulcld 8009 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℂ) |
8 | fvexg 5553 | . . 3 ⊢ ((√ ∈ V ∧ (𝐴 · (∗‘𝐴)) ∈ ℂ) → (√‘(𝐴 · (∗‘𝐴))) ∈ V) | |
9 | 4, 7, 8 | sylancr 414 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ V) |
10 | fveq2 5534 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
11 | oveq12 5906 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) | |
12 | 10, 11 | mpdan 421 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) |
13 | 12 | fveq2d 5538 | . . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴)))) |
14 | df-abs 11043 | . . 3 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
15 | 13, 14 | fvmptg 5613 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (√‘(𝐴 · (∗‘𝐴))) ∈ V) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
16 | 9, 15 | mpdan 421 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 class class class wbr 4018 ↦ cmpt 4079 ‘cfv 5235 ℩crio 5851 (class class class)co 5897 ℂcc 7840 ℝcr 7841 0cc0 7842 · cmul 7847 ≤ cle 8024 2c2 9001 ↑cexp 10553 ∗ccj 10883 √csqrt 11040 abscabs 11041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-ltxr 8028 df-sub 8161 df-neg 8162 df-reap 8563 df-cj 10886 df-rsqrt 11042 df-abs 11043 |
This theorem is referenced by: absneg 11094 abscl 11095 abscj 11096 absvalsq 11097 absval2 11101 abs0 11102 absi 11103 absge0 11104 absrpclap 11105 absmul 11113 absid 11115 absre 11121 absf 11154 |
Copyright terms: Public domain | W3C validator |