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Theorem eflt 15464

Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)

**Assertion**
Ref	Expression
eflt	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵)))

Proof of Theorem eflt Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efltim 12224	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
2		efcn 15457	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
3		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℝ)
4	3	recnd 8186	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℂ)
5		simpr 110	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) < (exp‘𝐵))
6		simpll 527	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐴 ∈ ℝ)
7	6	reefcld 12195	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) ∈ ℝ)
8	3	reefcld 12195	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐵) ∈ ℝ)
9		difrp 9900	. . . . . . 7 ⊢ (((exp‘𝐴) ∈ ℝ ∧ (exp‘𝐵) ∈ ℝ) → ((exp‘𝐴) < (exp‘𝐵) ↔ ((exp‘𝐵) − (exp‘𝐴)) ∈ ℝ⁺))
10	7, 8, 9	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → ((exp‘𝐴) < (exp‘𝐵) ↔ ((exp‘𝐵) − (exp‘𝐴)) ∈ ℝ⁺))
11	5, 10	mpbid 147	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → ((exp‘𝐵) − (exp‘𝐴)) ∈ ℝ⁺)
12		cncfi 15267	. . . . 5 ⊢ ((exp ∈ (ℂ–cn→ℂ) ∧ 𝐵 ∈ ℂ ∧ ((exp‘𝐵) − (exp‘𝐴)) ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))
13	2, 4, 11, 12	mp3an2i 1376	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))
14	6	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → 𝐴 ∈ ℝ)
15	3	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → 𝐵 ∈ ℝ)
16		simplr 528	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → (exp‘𝐴) < (exp‘𝐵))
17		simprl 529	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → 𝑑 ∈ ℝ⁺)
18		fvoveq1 6030	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 − 𝐵)) = (abs‘(𝐴 − 𝐵)))
19	18	breq1d 4093	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((abs‘(𝑥 − 𝐵)) < 𝑑 ↔ (abs‘(𝐴 − 𝐵)) < 𝑑))
20	19	imbrov2fvoveq 6032	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))) ↔ ((abs‘(𝐴 − 𝐵)) < 𝑑 → (abs‘((exp‘𝐴) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴)))))
21		simprr 531	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))
22	14	recnd 8186	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → 𝐴 ∈ ℂ)
23	20, 21, 22	rspcdva 2912	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → ((abs‘(𝐴 − 𝐵)) < 𝑑 → (abs‘((exp‘𝐴) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))
24	14, 15, 16, 17, 23	efltlemlt 15463	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) ∧ (𝑑 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ ((abs‘(𝑥 − 𝐵)) < 𝑑 → (abs‘((exp‘𝑥) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))) → 𝐴 < 𝐵)
25	13, 24	rexlimddv 2653	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐴 < 𝐵)
26	25	ex 115	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) < (exp‘𝐵) → 𝐴 < 𝐵))
27	1, 26	impbid 129	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 ℂcc 8008 ℝcr 8009 < clt 8192 − cmin 8328 ℝ⁺crp 9861 abscabs 11523 expce 12168 –cn→ccncf 15259

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130 ax-addf 8132 ax-mulf 8133

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7162 df-inf 7163 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-xneg 9980 df-xadd 9981 df-ico 10102 df-fz 10217 df-fzo 10351 df-seqfrec 10682 df-exp 10773 df-fac 10960 df-bc 10982 df-ihash 11010 df-shft 11341 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-clim 11805 df-sumdc 11880 df-ef 12174 df-rest 13289 df-topgen 13308 df-psmet 14522 df-xmet 14523 df-met 14524 df-bl 14525 df-mopn 14526 df-top 14687 df-topon 14700 df-bases 14732 df-ntr 14785 df-cn 14877 df-cnp 14878 df-tx 14942 df-cncf 15260 df-limced 15345 df-dvap 15346

This theorem is referenced by: efle 15465 reefiso 15466 reapef 15467 logdivlti 15570 cxplt 15605 rpcxplt2 15608

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