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Theorem cxplt3 26585

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 764	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ⁺)
2		simprl 768	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ)
3	2	recnd 11243	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ)
4		cxprec 26571	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
5	1, 3, 4	syl2anc 583	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
6		simprr 770	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ)
7	6	recnd 11243	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ)
8		cxprec 26571	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
9	1, 7, 8	syl2anc 583	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
10	5, 9	breq12d 5154	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
11	1	rprecred 13030	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (1 / 𝐴) ∈ ℝ)
12		simplr 766	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 < 1)
13	1	reclt1d 13032	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 231	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 < (1 / 𝐴))
15		cxplt 26579	. . 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 836	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
17		rpcxpcl 26561	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
18	17	ad2ant2rl 746	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
19		rpcxpcl 26561	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
20	19	ad2ant2r 744	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
21	18, 20	ltrecd 13037	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
22	10, 16, 21	3bitr4d 311	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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 ℂcc 11107 ℝcr 11108 1c1 11110 < clt 11249 / cdiv 11872 ℝ⁺crp 12977 ↑_𝑐ccxp 26440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-log 26441 df-cxp 26442

This theorem is referenced by: cxple3 26586 cxplt3d 26620

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