Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > exp1d | Structured version Visualization version GIF version |
Description: Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
expcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
exp1d | ⊢ (𝜑 → (𝐴↑1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | exp1 13797 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴↑1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 (class class class)co 7284 ℂcc 10878 1c1 10881 ↑cexp 13791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-er 8507 df-en 8743 df-dom 8744 df-sdom 8745 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-n0 12243 df-z 12329 df-uz 12592 df-seq 13731 df-exp 13792 |
This theorem is referenced by: faclbnd4lem1 14016 fsumcube 15779 sin01gt0 15908 rplpwr 16276 prmdvdsexp 16429 phiprm 16487 eulerthlem2 16492 pcelnn 16580 expnprm 16612 prmpwdvds 16614 pockthg 16616 odcau 19218 plyco 25411 dgrcolem1 25443 vieta1 25481 taylthlem1 25541 ftalem2 26232 vmaprm 26275 vma1 26324 1sgmprm 26356 chtublem 26368 fsumvma2 26371 chpchtsum 26376 logfacrlim2 26383 bposlem2 26442 bposlem6 26446 lgsval2lem 26464 2sqblem 26588 chebbnd1lem1 26626 rplogsumlem2 26642 rpvmasumlem 26644 ostth3 26795 nn0prpwlem 34520 nn0prpw 34521 bfplem1 35989 dvrelogpow2b 40083 aks4d1p1p4 40086 aks4d1p1p7 40089 aks4d1p1p5 40090 aks4d1p1 40091 aks4d1p3 40093 aks4d1p8d2 40100 2ap1caineq 40108 fltnltalem 40506 fltnlta 40507 3cubeslem3r 40516 rmxy1 40751 jm2.18 40817 jm2.23 40825 jm3.1lem2 40847 areaquad 41054 radcnvrat 41939 stoweidlem3 43551 wallispilem2 43614 stirlinglem1 43622 stirlinglem7 43628 stirlinglem10 43631 lighneal 45074 blenpw2m1 45936 |
Copyright terms: Public domain | W3C validator |