cxplt3 - Intuitionistic Logic Explorer

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Theorem cxplt3 15559

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 8143	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ)
4		cxprec 15549	. . . 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 8143	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ)
8		cxprec 15549	. . . 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 4075	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶) ↔ (1 / (𝐴↑_𝑐𝐵)) < (1 / (𝐴↑_𝑐𝐶))))
11	1	rprecred 9872	. . 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 9874	. . . 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 15555	. . 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 1253	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ ((1 / 𝐴)↑_𝑐𝐵) < ((1 / 𝐴)↑_𝑐𝐶)))
17		rpcxpcl 15542	. . . 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 15542	. . . 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 9879	. 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 1375 ∈ wcel 2180 class class class wbr 4062 (class class class)co 5974 ℂcc 7965 ℝcr 7966 1c1 7968 < clt 8149 / cdiv 8787 ℝ⁺crp 9817 ↑_𝑐ccxp 15496

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087 ax-pre-suploc 8088 ax-addf 8089 ax-mulf 8090

This theorem depends on definitions: df-bi 117 df-stab 835 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-disj 4039 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-of 6188 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-frec 6507 df-1o 6532 df-oadd 6536 df-er 6650 df-map 6767 df-pm 6768 df-en 6858 df-dom 6859 df-fin 6860 df-sup 7119 df-inf 7120 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-xneg 9936 df-xadd 9937 df-ioo 10056 df-ico 10058 df-icc 10059 df-fz 10173 df-fzo 10307 df-seqfrec 10637 df-exp 10728 df-fac 10915 df-bc 10937 df-ihash 10965 df-shft 11292 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-clim 11756 df-sumdc 11831 df-ef 12125 df-e 12126 df-rest 13240 df-topgen 13259 df-psmet 14472 df-xmet 14473 df-met 14474 df-bl 14475 df-mopn 14476 df-top 14637 df-topon 14650 df-bases 14682 df-ntr 14735 df-cn 14827 df-cnp 14828 df-tx 14892 df-cncf 15210 df-limced 15295 df-dvap 15296 df-relog 15497 df-rpcxp 15498

This theorem is referenced by: cxple3 15560 cxplt3d 15574

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