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Theorem cxplt3 24969

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

**Assertion**
Ref	Expression
cxplt3	⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵)))

**Proof of Theorem cxplt3**
Step	Hyp	Ref	Expression
1		simpll 763	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ⁺)
2		simprl 767	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ)
3	2	recnd 10520	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ)
4		cxprec 24955	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
5	1, 3, 4	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
6		simprr 769	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ)
7	6	recnd 10520	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ)
8		cxprec 24955	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
9	1, 7, 8	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
10	5, 9	breq12d 4979	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
11	1	rprecred 12297	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (1 / 𝐴) ∈ ℝ)
12		simplr 765	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 < 1)
13	1	reclt1d 12299	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 233	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 < (1 / 𝐴))
15		cxplt 24963	. . 3 ⊢ ((((1 / 𝐴) ∈ ℝ ∧ 1 < (1 / 𝐴)) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
16	11, 14, 2, 6, 15	syl22anc 835	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
17		rpcxpcl 24945	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
18	17	ad2ant2rl 745	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
19		rpcxpcl 24945	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
20	19	ad2ant2r 743	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
21	18, 20	ltrecd 12304	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
22	10, 16, 21	3bitr4d 312	1 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 class class class wbr 4966 (class class class)co 7021 ℂcc 10386 ℝcr 10387 1c1 10389 < clt 10526 / cdiv 11150 ℝ⁺crp 12244 ↑_𝑐ccxp 24825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-fi 8726 df-sup 8757 df-inf 8758 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-q 12203 df-rp 12245 df-xneg 12362 df-xadd 12363 df-xmul 12364 df-ioo 12597 df-ioc 12598 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-fl 13017 df-mod 13093 df-seq 13225 df-exp 13285 df-fac 13489 df-bc 13518 df-hash 13546 df-shft 14265 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-limsup 14667 df-clim 14684 df-rlim 14685 df-sum 14882 df-ef 15259 df-sin 15261 df-cos 15262 df-pi 15264 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-rest 16530 df-topn 16531 df-0g 16549 df-gsum 16550 df-topgen 16551 df-pt 16552 df-prds 16555 df-xrs 16609 df-qtop 16614 df-imas 16615 df-xps 16617 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-mulg 17987 df-cntz 18193 df-cmn 18640 df-psmet 20224 df-xmet 20225 df-met 20226 df-bl 20227 df-mopn 20228 df-fbas 20229 df-fg 20230 df-cnfld 20233 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-lp 21433 df-perf 21434 df-cn 21524 df-cnp 21525 df-haus 21612 df-tx 21859 df-hmeo 22052 df-fil 22143 df-fm 22235 df-flim 22236 df-flf 22237 df-xms 22618 df-ms 22619 df-tms 22620 df-cncf 23174 df-limc 24152 df-dv 24153 df-log 24826 df-cxp 24827

This theorem is referenced by: cxple3 24970 cxplt3d 25003

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