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Mirrors > Home > MPE Home > Th. List > absval | Structured version Visualization version 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 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
2 | oveq12 7154 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) | |
3 | 1, 2 | mpdan 683 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴))) |
4 | 3 | fveq2d 6667 | . 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴)))) |
5 | df-abs 14583 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
6 | fvex 6676 | . 2 ⊢ (√‘(𝐴 · (∗‘𝐴))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6761 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 · cmul 10530 ∗ccj 14443 √csqrt 14580 abscabs 14581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-abs 14583 |
This theorem is referenced by: absneg 14625 abscl 14626 abscj 14627 absvalsq 14628 absval2 14632 abs0 14633 absi 14634 absge0 14635 absrpcl 14636 absmul 14642 absid 14644 absre 14649 absf 14685 cphabscl 23716 cphipipcj 23731 tcphcphlem2 23766 siii 28557 norm-iii-i 28843 absfico 41357 |
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