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Mirrors > Home > MPE Home > Th. List > cxple2a | Structured version Visualization version GIF version |
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
cxple2a | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1189 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 𝐴 ≤ 𝐵) | |
2 | simp11 1199 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 𝐴 ∈ ℝ) |
4 | simpl2l 1222 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 0 ≤ 𝐴) | |
5 | simp12 1200 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) | |
6 | 5 | adantr 483 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 𝐵 ∈ ℝ) |
7 | 0red 10630 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 0 ∈ ℝ) | |
8 | 7, 3, 6, 4, 1 | letrd 10783 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 0 ≤ 𝐵) |
9 | simp13 1201 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 𝐶 ∈ ℝ) | |
10 | 9 | anim1i 616 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
11 | elrp 12378 | . . . . 5 ⊢ (𝐶 ∈ ℝ+ ↔ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) | |
12 | 10, 11 | sylibr 236 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → 𝐶 ∈ ℝ+) |
13 | cxple2 25266 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶))) | |
14 | 3, 4, 6, 8, 12, 13 | syl221anc 1377 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → (𝐴 ≤ 𝐵 ↔ (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶))) |
15 | 1, 14 | mpbid 234 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 < 𝐶) → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)) |
16 | 1le1 11254 | . . . 4 ⊢ 1 ≤ 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 = 𝐶) → 1 ≤ 1) |
18 | 2 | recnd 10655 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℂ) |
19 | cxp0 25239 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑𝑐0) = 1) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (𝐴↑𝑐0) = 1) |
21 | oveq2 7150 | . . . 4 ⊢ (0 = 𝐶 → (𝐴↑𝑐0) = (𝐴↑𝑐𝐶)) | |
22 | 20, 21 | sylan9req 2877 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 = 𝐶) → 1 = (𝐴↑𝑐𝐶)) |
23 | 5 | recnd 10655 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℂ) |
24 | cxp0 25239 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵↑𝑐0) = 1) | |
25 | 23, 24 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (𝐵↑𝑐0) = 1) |
26 | oveq2 7150 | . . . 4 ⊢ (0 = 𝐶 → (𝐵↑𝑐0) = (𝐵↑𝑐𝐶)) | |
27 | 25, 26 | sylan9req 2877 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 = 𝐶) → 1 = (𝐵↑𝑐𝐶)) |
28 | 17, 22, 27 | 3brtr3d 5083 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) ∧ 0 = 𝐶) → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)) |
29 | simp2r 1196 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → 0 ≤ 𝐶) | |
30 | 0re 10629 | . . . 4 ⊢ 0 ∈ ℝ | |
31 | leloe 10713 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (0 ≤ 𝐶 ↔ (0 < 𝐶 ∨ 0 = 𝐶))) | |
32 | 30, 9, 31 | sylancr 589 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (0 ≤ 𝐶 ↔ (0 < 𝐶 ∨ 0 = 𝐶))) |
33 | 29, 32 | mpbid 234 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (0 < 𝐶 ∨ 0 = 𝐶)) |
34 | 15, 28, 33 | mpjaodan 955 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5052 (class class class)co 7142 ℂcc 10521 ℝcr 10522 0cc0 10523 1c1 10524 < clt 10661 ≤ cle 10662 ℝ+crp 12376 ↑𝑐ccxp 25125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ioc 12730 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14411 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-limsup 14813 df-clim 14830 df-rlim 14831 df-sum 15028 df-ef 15406 df-sin 15408 df-cos 15409 df-pi 15411 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 df-perf 21728 df-cn 21818 df-cnp 21819 df-haus 21906 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-xms 22913 df-ms 22914 df-tms 22915 df-cncf 23469 df-limc 24449 df-dv 24450 df-log 25126 df-cxp 25127 |
This theorem is referenced by: cxple2ad 25294 |
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