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Mirrors > Home > MPE Home > Th. List > absef | Structured version Visualization version GIF version |
Description: The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
Ref | Expression |
---|---|
absef | ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | replim 13704 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
2 | 1 | fveq2d 6107 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴))))) |
3 | recl 13698 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
4 | 3 | recnd 9947 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
5 | ax-icn 9874 | . . . . . . 7 ⊢ i ∈ ℂ | |
6 | imcl 13699 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
7 | 6 | recnd 9947 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
8 | mulcl 9899 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
9 | 5, 7, 8 | sylancr 694 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
10 | efadd 14663 | . . . . . 6 ⊢ (((ℜ‘𝐴) ∈ ℂ ∧ (i · (ℑ‘𝐴)) ∈ ℂ) → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) | |
11 | 4, 9, 10 | syl2anc 691 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
12 | 2, 11 | eqtrd 2644 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
13 | 12 | fveq2d 6107 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴)))))) |
14 | 3 | reefcld 14657 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℝ) |
15 | 14 | recnd 9947 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℂ) |
16 | efcl 14652 | . . . . 5 ⊢ ((i · (ℑ‘𝐴)) ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) | |
17 | 9, 16 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) |
18 | 15, 17 | absmuld 14041 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴)))))) |
19 | absefi 14765 | . . . . 5 ⊢ ((ℑ‘𝐴) ∈ ℝ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) | |
20 | 6, 19 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) |
21 | 20 | oveq2d 6565 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
22 | 13, 18, 21 | 3eqtrd 2648 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
23 | 15 | abscld 14023 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℝ) |
24 | 23 | recnd 9947 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℂ) |
25 | 24 | mulid1d 9936 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · 1) = (abs‘(exp‘(ℜ‘𝐴)))) |
26 | efgt0 14672 | . . . . 5 ⊢ ((ℜ‘𝐴) ∈ ℝ → 0 < (exp‘(ℜ‘𝐴))) | |
27 | 3, 26 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 < (exp‘(ℜ‘𝐴))) |
28 | 0re 9919 | . . . . 5 ⊢ 0 ∈ ℝ | |
29 | ltle 10005 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ (exp‘(ℜ‘𝐴)) ∈ ℝ) → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) | |
30 | 28, 14, 29 | sylancr 694 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) |
31 | 27, 30 | mpd 15 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (exp‘(ℜ‘𝐴))) |
32 | 14, 31 | absidd 14009 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) = (exp‘(ℜ‘𝐴))) |
33 | 22, 25, 32 | 3eqtrd 2648 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 ℜcre 13685 ℑcim 13686 abscabs 13822 expce 14631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 |
This theorem is referenced by: absefib 14767 eff1olem 24098 relog 24147 abscxp 24238 abscxp2 24239 abscxpbnd 24294 zetacvg 24541 |
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