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Mirrors > Home > MPE Home > Th. List > cxple2a | Structured version Unicode version |
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
cxple2a |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 993 |
. . 3
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2 | simp11 1018 |
. . . . 5
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3 | 2 | adantr 465 |
. . . 4
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4 | simpl2l 1041 |
. . . 4
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5 | simp12 1019 |
. . . . 5
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6 | 5 | adantr 465 |
. . . 4
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7 | 0red 9499 |
. . . . 5
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8 | 7, 3, 6, 4, 1 | letrd 9640 |
. . . 4
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9 | simp13 1020 |
. . . . . 6
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10 | 9 | anim1i 568 |
. . . . 5
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11 | elrp 11105 |
. . . . 5
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12 | 10, 11 | sylibr 212 |
. . . 4
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13 | cxple2 22276 |
. . . 4
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14 | 3, 4, 6, 8, 12, 13 | syl221anc 1230 |
. . 3
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15 | 1, 14 | mpbid 210 |
. 2
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16 | 1le1 10076 |
. . . 4
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17 | 16 | a1i 11 |
. . 3
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18 | 2 | recnd 9524 |
. . . . 5
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19 | cxp0 22249 |
. . . . 5
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20 | 18, 19 | syl 16 |
. . . 4
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21 | oveq2 6209 |
. . . 4
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22 | 20, 21 | sylan9req 2516 |
. . 3
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23 | 5 | recnd 9524 |
. . . . 5
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24 | cxp0 22249 |
. . . . 5
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25 | 23, 24 | syl 16 |
. . . 4
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26 | oveq2 6209 |
. . . 4
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27 | 25, 26 | sylan9req 2516 |
. . 3
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28 | 17, 22, 27 | 3brtr3d 4430 |
. 2
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29 | simp2r 1015 |
. . 3
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30 | 0re 9498 |
. . . 4
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31 | leloe 9573 |
. . . 4
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32 | 30, 9, 31 | sylancr 663 |
. . 3
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33 | 29, 32 | mpbid 210 |
. 2
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34 | 15, 28, 33 | mpjaodan 784 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-inf2 7959 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 ax-pre-sup 9472 ax-addf 9473 ax-mulf 9474 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-iin 4283 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-se 4789 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-isom 5536 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-of 6431 df-om 6588 df-1st 6688 df-2nd 6689 df-supp 6802 df-recs 6943 df-rdg 6977 df-1o 7031 df-2o 7032 df-oadd 7035 df-er 7212 df-map 7327 df-pm 7328 df-ixp 7375 df-en 7422 df-dom 7423 df-sdom 7424 df-fin 7425 df-fsupp 7733 df-fi 7773 df-sup 7803 df-oi 7836 df-card 8221 df-cda 8449 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-div 10106 df-nn 10435 df-2 10492 df-3 10493 df-4 10494 df-5 10495 df-6 10496 df-7 10497 df-8 10498 df-9 10499 df-10 10500 df-n0 10692 df-z 10759 df-dec 10868 df-uz 10974 df-q 11066 df-rp 11104 df-xneg 11201 df-xadd 11202 df-xmul 11203 df-ioo 11416 df-ioc 11417 df-ico 11418 df-icc 11419 df-fz 11556 df-fzo 11667 df-fl 11760 df-mod 11827 df-seq 11925 df-exp 11984 df-fac 12170 df-bc 12197 df-hash 12222 df-shft 12675 df-cj 12707 df-re 12708 df-im 12709 df-sqr 12843 df-abs 12844 df-limsup 13068 df-clim 13085 df-rlim 13086 df-sum 13283 df-ef 13472 df-sin 13474 df-cos 13475 df-pi 13477 df-struct 14295 df-ndx 14296 df-slot 14297 df-base 14298 df-sets 14299 df-ress 14300 df-plusg 14371 df-mulr 14372 df-starv 14373 df-sca 14374 df-vsca 14375 df-ip 14376 df-tset 14377 df-ple 14378 df-ds 14380 df-unif 14381 df-hom 14382 df-cco 14383 df-rest 14481 df-topn 14482 df-0g 14500 df-gsum 14501 df-topgen 14502 df-pt 14503 df-prds 14506 df-xrs 14560 df-qtop 14565 df-imas 14566 df-xps 14568 df-mre 14644 df-mrc 14645 df-acs 14647 df-mnd 15535 df-submnd 15585 df-mulg 15668 df-cntz 15955 df-cmn 16401 df-psmet 17935 df-xmet 17936 df-met 17937 df-bl 17938 df-mopn 17939 df-fbas 17940 df-fg 17941 df-cnfld 17945 df-top 18636 df-bases 18638 df-topon 18639 df-topsp 18640 df-cld 18756 df-ntr 18757 df-cls 18758 df-nei 18835 df-lp 18873 df-perf 18874 df-cn 18964 df-cnp 18965 df-haus 19052 df-tx 19268 df-hmeo 19461 df-fil 19552 df-fm 19644 df-flim 19645 df-flf 19646 df-xms 20028 df-ms 20029 df-tms 20030 df-cncf 20587 df-limc 21475 df-dv 21476 df-log 22142 df-cxp 22143 |
This theorem is referenced by: cxple2ad 22304 |
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