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Mirrors > Home > MPE Home > Th. List > abscj | Structured version Visualization version GIF version |
Description: The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.) |
Ref | Expression |
---|---|
abscj | ⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjcl 14463 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
2 | absval 14596 | . . . 4 ⊢ ((∗‘𝐴) ∈ ℂ → (abs‘(∗‘𝐴)) = (√‘((∗‘𝐴) · (∗‘(∗‘𝐴))))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (√‘((∗‘𝐴) · (∗‘(∗‘𝐴))))) |
4 | mulcom 10622 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (𝐴 · (∗‘𝐴)) = ((∗‘𝐴) · 𝐴)) | |
5 | 1, 4 | mpdan 685 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) = ((∗‘𝐴) · 𝐴)) |
6 | cjcj 14498 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = 𝐴) | |
7 | 6 | oveq2d 7171 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((∗‘𝐴) · (∗‘(∗‘𝐴))) = ((∗‘𝐴) · 𝐴)) |
8 | 5, 7 | eqtr4d 2859 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) = ((∗‘𝐴) · (∗‘(∗‘𝐴)))) |
9 | 8 | fveq2d 6673 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) = (√‘((∗‘𝐴) · (∗‘(∗‘𝐴))))) |
10 | 3, 9 | eqtr4d 2859 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (√‘(𝐴 · (∗‘𝐴)))) |
11 | absval 14596 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
12 | 10, 11 | eqtr4d 2859 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 · cmul 10541 ∗ccj 14454 √csqrt 14591 abscabs 14592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-2 11699 df-cj 14457 df-re 14458 df-im 14459 df-abs 14594 |
This theorem is referenced by: abstri 14689 abs1m 14694 abscji 14760 abscjd 14809 |
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