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Mirrors > Home > ILE Home > Th. List > cxplt | GIF version |
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
cxplt | ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 521 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ) | |
2 | rplogcl 13139 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+) | |
3 | 2 | adantr 274 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (log‘𝐴) ∈ ℝ+) |
4 | 3 | rpred 9581 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (log‘𝐴) ∈ ℝ) |
5 | 1, 4 | remulcld 7887 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 · (log‘𝐴)) ∈ ℝ) |
6 | simprr 522 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ) | |
7 | 6, 4 | remulcld 7887 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐶 · (log‘𝐴)) ∈ ℝ) |
8 | eflt 13035 | . . 3 ⊢ (((𝐵 · (log‘𝐴)) ∈ ℝ ∧ (𝐶 · (log‘𝐴)) ∈ ℝ) → ((𝐵 · (log‘𝐴)) < (𝐶 · (log‘𝐴)) ↔ (exp‘(𝐵 · (log‘𝐴))) < (exp‘(𝐶 · (log‘𝐴))))) | |
9 | 5, 7, 8 | syl2anc 409 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐵 · (log‘𝐴)) < (𝐶 · (log‘𝐴)) ↔ (exp‘(𝐵 · (log‘𝐴))) < (exp‘(𝐶 · (log‘𝐴))))) |
10 | 1, 6, 3 | ltmul1d 9623 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐵 · (log‘𝐴)) < (𝐶 · (log‘𝐴)))) |
11 | simpll 519 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ) | |
12 | 0red 7858 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 0 ∈ ℝ) | |
13 | 1red 7872 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 ∈ ℝ) | |
14 | 0lt1 7981 | . . . . . . 7 ⊢ 0 < 1 | |
15 | 14 | a1i 9 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 0 < 1) |
16 | simplr 520 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 1 < 𝐴) | |
17 | 12, 13, 11, 15, 16 | lttrd 7980 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 0 < 𝐴) |
18 | 11, 17 | elrpd 9578 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ+) |
19 | 1 | recnd 7885 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℂ) |
20 | rpcxpef 13154 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
21 | 18, 19, 20 | syl2anc 409 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
22 | 6 | recnd 7885 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℂ) |
23 | rpcxpef 13154 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴)))) | |
24 | 18, 22, 23 | syl2anc 409 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴)))) |
25 | 21, 24 | breq12d 3974 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶) ↔ (exp‘(𝐵 · (log‘𝐴))) < (exp‘(𝐶 · (log‘𝐴))))) |
26 | 9, 10, 25 | 3bitr4d 219 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 2125 class class class wbr 3961 ‘cfv 5163 (class class class)co 5814 ℂcc 7709 ℝcr 7710 0cc0 7711 1c1 7712 · cmul 7716 < clt 7891 ℝ+crp 9538 expce 11516 logclog 13116 ↑𝑐ccxp 13117 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 ax-pre-suploc 7832 ax-addf 7833 ax-mulf 7834 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-disj 3939 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-of 6022 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-oadd 6357 df-er 6469 df-map 6584 df-pm 6585 df-en 6675 df-dom 6676 df-fin 6677 df-sup 6916 df-inf 6917 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-xneg 9657 df-xadd 9658 df-ioo 9774 df-ico 9776 df-icc 9777 df-fz 9891 df-fzo 10020 df-seqfrec 10323 df-exp 10397 df-fac 10577 df-bc 10599 df-ihash 10627 df-shft 10692 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-sumdc 11228 df-ef 11522 df-e 11523 df-rest 12292 df-topgen 12311 df-psmet 12326 df-xmet 12327 df-met 12328 df-bl 12329 df-mopn 12330 df-top 12335 df-topon 12348 df-bases 12380 df-ntr 12435 df-cn 12527 df-cnp 12528 df-tx 12592 df-cncf 12897 df-limced 12964 df-dvap 12965 df-relog 13118 df-rpcxp 13119 |
This theorem is referenced by: cxple 13176 cxplt3 13179 cxpltd 13187 |
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