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Mirrors > Home > MPE Home > Th. List > absefi | Structured version Visualization version GIF version |
Description: The absolute value of the exponential function of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.) |
Ref | Expression |
---|---|
absefi | ⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10232 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | efival 15088 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴)))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴)))) |
4 | 3 | fveq2d 6337 | . 2 ⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = (abs‘((cos‘𝐴) + (i · (sin‘𝐴))))) |
5 | recoscl 15077 | . . . 4 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | |
6 | resincl 15076 | . . . 4 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | |
7 | absreim 14241 | . . . 4 ⊢ (((cos‘𝐴) ∈ ℝ ∧ (sin‘𝐴) ∈ ℝ) → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)))) | |
8 | 5, 6, 7 | syl2anc 573 | . . 3 ⊢ (𝐴 ∈ ℝ → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)))) |
9 | 5 | resqcld 13242 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴)↑2) ∈ ℝ) |
10 | 9 | recnd 10274 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴)↑2) ∈ ℂ) |
11 | 6 | resqcld 13242 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴)↑2) ∈ ℝ) |
12 | 11 | recnd 10274 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴)↑2) ∈ ℂ) |
13 | 10, 12 | addcomd 10444 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
14 | sincossq 15112 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) | |
15 | 1, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
16 | 13, 15 | eqtrd 2805 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = 1) |
17 | 16 | fveq2d 6337 | . . . 4 ⊢ (𝐴 ∈ ℝ → (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) = (√‘1)) |
18 | sqrt1 14220 | . . . 4 ⊢ (√‘1) = 1 | |
19 | 17, 18 | syl6eq 2821 | . . 3 ⊢ (𝐴 ∈ ℝ → (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) = 1) |
20 | 8, 19 | eqtrd 2805 | . 2 ⊢ (𝐴 ∈ ℝ → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = 1) |
21 | 4, 20 | eqtrd 2805 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 ℂcc 10140 ℝcr 10141 1c1 10143 ici 10144 + caddc 10145 · cmul 10147 2c2 11276 ↑cexp 13067 √csqrt 14181 abscabs 14182 expce 14998 sincsin 15000 cosccos 15001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-rp 12036 df-ico 12386 df-fz 12534 df-fzo 12674 df-fl 12801 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 |
This theorem is referenced by: absef 15133 efieq1re 15135 pige3 24490 efif1olem4 24512 efifo 24514 |
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