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Mirrors > Home > MPE Home > Th. List > absefi | Structured version Visualization version GIF version |
Description: The absolute value of the exponential 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 10616 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | efival 15497 | . . . 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 6649 | . 2 ⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = (abs‘((cos‘𝐴) + (i · (sin‘𝐴))))) |
5 | recoscl 15486 | . . . 4 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | |
6 | resincl 15485 | . . . 4 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | |
7 | absreim 14645 | . . . 4 ⊢ (((cos‘𝐴) ∈ ℝ ∧ (sin‘𝐴) ∈ ℝ) → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)))) | |
8 | 5, 6, 7 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ ℝ → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)))) |
9 | 5 | resqcld 13607 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴)↑2) ∈ ℝ) |
10 | 9 | recnd 10658 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴)↑2) ∈ ℂ) |
11 | 6 | resqcld 13607 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴)↑2) ∈ ℝ) |
12 | 11 | recnd 10658 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴)↑2) ∈ ℂ) |
13 | 10, 12 | addcomd 10831 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
14 | sincossq 15521 | . . . . . . 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 2833 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = 1) |
17 | 16 | fveq2d 6649 | . . . 4 ⊢ (𝐴 ∈ ℝ → (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) = (√‘1)) |
18 | sqrt1 14623 | . . . 4 ⊢ (√‘1) = 1 | |
19 | 17, 18 | eqtrdi 2849 | . . 3 ⊢ (𝐴 ∈ ℝ → (√‘(((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) = 1) |
20 | 8, 19 | eqtrd 2833 | . 2 ⊢ (𝐴 ∈ ℝ → (abs‘((cos‘𝐴) + (i · (sin‘𝐴)))) = 1) |
21 | 4, 20 | eqtrd 2833 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 2c2 11680 ↑cexp 13425 √csqrt 14584 abscabs 14585 expce 15407 sincsin 15409 cosccos 15410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 |
This theorem is referenced by: absef 15542 efieq1re 15544 pige3ALT 25112 efif1olem4 25137 efifo 25139 |
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