cxplt3 - Intuitionistic Logic Explorer

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Theorem cxplt3 14425

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 527	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ⁺)
2		simprl 529	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ)
3	2	recnd 7988	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ)
4		cxprec 14416	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
5	1, 3, 4	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐵) = (1 / (𝐴↑_𝑐𝐵)))
6		simprr 531	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ)
7	6	recnd 7988	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ)
8		cxprec 14416	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
9	1, 7, 8	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((1 / 𝐴)↑_𝑐𝐶) = (1 / (𝐴↑_𝑐𝐶)))
10	5, 9	breq12d 4018	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
11	1	rprecred 9710	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (1 / 𝐴) ∈ ℝ)
12		simplr 528	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 < 1)
13	1	reclt1d 9712	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 147	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 < (1 / 𝐴))
15		cxplt 14421	. . 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 1239	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
17		rpcxpcl 14409	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
18	17	ad2ant2rl 511	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
19		rpcxpcl 14409	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
20	19	ad2ant2r 509	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
21	18, 20	ltrecd 9717	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐴↑_𝑐𝐶) < (𝐴↑_𝑐𝐵) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
22	10, 16, 21	3bitr4d 220	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 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 (class class class)co 5877 ℂcc 7811 ℝcr 7812 1c1 7814 < clt 7994 / cdiv 8631 ℝ⁺crp 9655 ↑_𝑐ccxp 14363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-pre-suploc 7934 ax-addf 7935 ax-mulf 7936

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-map 6652 df-pm 6653 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-ioo 9894 df-ico 9896 df-icc 9897 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-shft 10826 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-e 11659 df-rest 12695 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-met 13534 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-bases 13628 df-ntr 13681 df-cn 13773 df-cnp 13774 df-tx 13838 df-cncf 14143 df-limced 14210 df-dvap 14211 df-relog 14364 df-rpcxp 14365

This theorem is referenced by: cxple3 14426 cxplt3d 14439

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