![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > cxple2ad | Unicode version |
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
recxpcld.1 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
recxpcld.2 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
recxpcld.3 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
cxple2ad.4 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
cxple2ad.5 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
cxple2ad.6 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
cxple2ad |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recxpcld.1 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | recxpcld.3 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | cxple2ad.4 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | recxpcld.2 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | cxple2ad.5 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | cxple2ad.6 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | cxple2a 20547 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 1, 2, 3, 4, 5, 6, 7 | syl321anc 1206 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: abscxpbnd 20594 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-inf2 7556 ax-cnex 9006 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 ax-pre-mulgt0 9027 ax-pre-sup 9028 ax-addf 9029 ax-mulf 9030 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-int 4015 df-iun 4059 df-iin 4060 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-se 4506 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-om 4809 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-isom 5426 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-of 6268 df-1st 6312 df-2nd 6313 df-riota 6512 df-recs 6596 df-rdg 6631 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6868 df-map 6983 df-pm 6984 df-ixp 7027 df-en 7073 df-dom 7074 df-sdom 7075 df-fin 7076 df-fi 7378 df-sup 7408 df-oi 7439 df-card 7786 df-cda 8008 df-pnf 9082 df-mnf 9083 df-xr 9084 df-ltxr 9085 df-le 9086 df-sub 9253 df-neg 9254 df-div 9638 df-nn 9961 df-2 10018 df-3 10019 df-4 10020 df-5 10021 df-6 10022 df-7 10023 df-8 10024 df-9 10025 df-10 10026 df-n0 10182 df-z 10243 df-dec 10343 df-uz 10449 df-q 10535 df-rp 10573 df-xneg 10670 df-xadd 10671 df-xmul 10672 df-ioo 10880 df-ioc 10881 df-ico 10882 df-icc 10883 df-fz 11004 df-fzo 11095 df-fl 11161 df-mod 11210 df-seq 11283 df-exp 11342 df-fac 11526 df-bc 11553 df-hash 11578 df-shft 11841 df-cj 11863 df-re 11864 df-im 11865 df-sqr 11999 df-abs 12000 df-limsup 12224 df-clim 12241 df-rlim 12242 df-sum 12439 df-ef 12629 df-sin 12631 df-cos 12632 df-pi 12634 df-struct 13430 df-ndx 13431 df-slot 13432 df-base 13433 df-sets 13434 df-ress 13435 df-plusg 13501 df-mulr 13502 df-starv 13503 df-sca 13504 df-vsca 13505 df-tset 13507 df-ple 13508 df-ds 13510 df-unif 13511 df-hom 13512 df-cco 13513 df-rest 13609 df-topn 13610 df-topgen 13626 df-pt 13627 df-prds 13630 df-xrs 13685 df-0g 13686 df-gsum 13687 df-qtop 13692 df-imas 13693 df-xps 13695 df-mre 13770 df-mrc 13771 df-acs 13773 df-mnd 14649 df-submnd 14698 df-mulg 14774 df-cntz 15075 df-cmn 15373 df-psmet 16653 df-xmet 16654 df-met 16655 df-bl 16656 df-mopn 16657 df-fbas 16658 df-fg 16659 df-cnfld 16663 df-top 16922 df-bases 16924 df-topon 16925 df-topsp 16926 df-cld 17042 df-ntr 17043 df-cls 17044 df-nei 17121 df-lp 17159 df-perf 17160 df-cn 17249 df-cnp 17250 df-haus 17337 df-tx 17551 df-hmeo 17744 df-fil 17835 df-fm 17927 df-flim 17928 df-flf 17929 df-xms 18307 df-ms 18308 df-tms 18309 df-cncf 18865 df-limc 19710 df-dv 19711 df-log 20411 df-cxp 20412 |
Copyright terms: Public domain | W3C validator |