cxplt3 - Intuitionistic Logic Explorer

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Theorem cxplt3 13480

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 519	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ⁺)
2		simprl 521	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ)
3	2	recnd 7927	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ)
4		cxprec 13471	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
5	1, 3, 4	syl2anc 409	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
6		simprr 522	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ)
7	6	recnd 7927	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ)
8		cxprec 13471	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
9	1, 7, 8	syl2anc 409	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
10	5, 9	breq12d 3995	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
11	1	rprecred 9644	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (1 / 𝐴) ∈ ℝ)
12		simplr 520	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 < 1)
13	1	reclt1d 9646	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 146	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 < (1 / 𝐴))
15		cxplt 13476	. . 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 1229	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
17		rpcxpcl 13464	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
18	17	ad2ant2rl 503	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
19		rpcxpcl 13464	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
20	19	ad2ant2r 501	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
21	18, 20	ltrecd 9651	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
22	10, 16, 21	3bitr4d 219	1 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1343 ∈ wcel 2136 class class class wbr 3982 (class class class)co 5842 ℂcc 7751 ℝcr 7752 1c1 7754 < clt 7933 / cdiv 8568 ℝ⁺crp 9589 ↑_𝑐ccxp 13418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-e 11590 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266 df-relog 13419 df-rpcxp 13420

This theorem is referenced by: cxple3 13481 cxplt3d 13494

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