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Mirrors > Home > MPE Home > Th. List > atanlogadd | Structured version Visualization version GIF version |
Description: The rule √(𝑧𝑤) = (√𝑧)(√𝑤) is not always true on the complex numbers, but it is true when the arguments of 𝑧 and 𝑤 sum to within the interval (-π, π], so there are some cases such as this one with 𝑧 = 1 + i𝐴 and 𝑤 = 1 − i𝐴 which are true unconditionally. This result can also be stated as "√(1 + 𝑧) + √(1 − 𝑧) is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
atanlogadd | ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10637 | . 2 ⊢ (𝐴 ∈ dom arctan → 0 ∈ ℝ) | |
2 | atandm2 25453 | . . . 4 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0)) | |
3 | 2 | simp1bi 1140 | . . 3 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ) |
4 | 3 | recld 14548 | . 2 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ) |
5 | atanlogaddlem 25489 | . 2 ⊢ ((𝐴 ∈ dom arctan ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) | |
6 | ax-1cn 10588 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
7 | ax-icn 10589 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | mulcl 10614 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
9 | 7, 3, 8 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ) |
10 | addcl 10612 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ) | |
11 | 6, 9, 10 | sylancr 589 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ) |
12 | 2 | simp3bi 1142 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0) |
13 | 11, 12 | logcld 25152 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ) |
14 | subcl 10878 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ) | |
15 | 6, 9, 14 | sylancr 589 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ) |
16 | 2 | simp2bi 1141 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0) |
17 | 15, 16 | logcld 25152 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ) |
18 | 13, 17 | addcomd 10835 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) + (log‘(1 + (i · 𝐴))))) |
19 | mulneg2 11070 | . . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
20 | 7, 3, 19 | sylancr 589 | . . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (i · -𝐴) = -(i · 𝐴)) |
21 | 20 | oveq2d 7165 | . . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴))) |
22 | negsub 10927 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + -(i · 𝐴)) = (1 − (i · 𝐴))) | |
23 | 6, 9, 22 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (1 + -(i · 𝐴)) = (1 − (i · 𝐴))) |
24 | 21, 23 | eqtrd 2855 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴))) |
25 | 24 | fveq2d 6667 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴)))) |
26 | 20 | oveq2d 7165 | . . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴))) |
27 | subneg 10928 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴))) | |
28 | 6, 9, 27 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (1 − -(i · 𝐴)) = (1 + (i · 𝐴))) |
29 | 26, 28 | eqtrd 2855 | . . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴))) |
30 | 29 | fveq2d 6667 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴)))) |
31 | 25, 30 | oveq12d 7167 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) + (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) + (log‘(1 + (i · 𝐴))))) |
32 | 18, 31 | eqtr4d 2858 | . . . 4 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) + (log‘(1 − (i · -𝐴))))) |
33 | 32 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) + (log‘(1 − (i · -𝐴))))) |
34 | atandmneg 25482 | . . . 4 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan) | |
35 | 4 | le0neg1d 11204 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) ≤ 0 ↔ 0 ≤ -(ℜ‘𝐴))) |
36 | 35 | biimpa 479 | . . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → 0 ≤ -(ℜ‘𝐴)) |
37 | 3 | renegd 14563 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
38 | 37 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
39 | 36, 38 | breqtrrd 5087 | . . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → 0 ≤ (ℜ‘-𝐴)) |
40 | atanlogaddlem 25489 | . . . 4 ⊢ ((-𝐴 ∈ dom arctan ∧ 0 ≤ (ℜ‘-𝐴)) → ((log‘(1 + (i · -𝐴))) + (log‘(1 − (i · -𝐴)))) ∈ ran log) | |
41 | 34, 39, 40 | syl2an2r 683 | . . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → ((log‘(1 + (i · -𝐴))) + (log‘(1 − (i · -𝐴)))) ∈ ran log) |
42 | 33, 41 | eqeltrd 2912 | . 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≤ 0) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) |
43 | 1, 4, 5, 42 | lecasei 10739 | 1 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 class class class wbr 5059 dom cdm 5548 ran crn 5549 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 0cc0 10530 1c1 10531 ici 10532 + caddc 10533 · cmul 10535 ≤ cle 10669 − cmin 10863 -cneg 10864 ℜcre 14451 logclog 25136 arctancatan 25440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-fac 13631 df-bc 13660 df-hash 13688 df-shft 14421 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-limsup 14823 df-clim 14840 df-rlim 14841 df-sum 15038 df-ef 15416 df-sin 15418 df-cos 15419 df-pi 15421 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-pt 16713 df-prds 16716 df-xrs 16770 df-qtop 16775 df-imas 16776 df-xps 16778 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-mulg 18220 df-cntz 18442 df-cmn 18903 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-fbas 20537 df-fg 20538 df-cnfld 20541 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-cld 21622 df-ntr 21623 df-cls 21624 df-nei 21701 df-lp 21739 df-perf 21740 df-cn 21830 df-cnp 21831 df-haus 21918 df-tx 22165 df-hmeo 22358 df-fil 22449 df-fm 22541 df-flim 22542 df-flf 22543 df-xms 22925 df-ms 22926 df-tms 22927 df-cncf 23481 df-limc 24461 df-dv 24462 df-log 25138 df-atan 25443 |
This theorem is referenced by: efiatan2 25493 |
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