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Theorem eflt 15640

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 12384	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
2		efcn 15633	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
3		simplr 529	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℝ)
4	3	recnd 8302	. . . . 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 12355	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) ∈ ℝ)
8	3	reefcld 12355	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐵) ∈ ℝ)
9		difrp 10025	. . . . . . 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 15443	. . . . 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 1379	. . . 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 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))))) → (exp‘𝐴) < (exp‘𝐵))
17		simprl 531	. . . . 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 6073	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 − 𝐵)) = (abs‘(𝐴 − 𝐵)))
19	18	breq1d 4119	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((abs‘(𝑥 − 𝐵)) < 𝑑 ↔ (abs‘(𝐴 − 𝐵)) < 𝑑))
20	19	imbrov2fvoveq 6075	. . . . . 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 533	. . . . . 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 8302	. . . . . 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 2926	. . . . 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 15639	. . . 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 2665	. . 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 1398 ∈ wcel 2203 ∀wral 2520 ∃wrex 2521 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 ℂcc 8125 ℝcr 8126 < clt 8308 − cmin 8444 ℝ⁺crp 9986 abscabs 11682 expce 12328 –cn→ccncf 15435

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247 ax-addf 8249 ax-mulf 8250

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-disj 4086 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-of 6266 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-oadd 6651 df-er 6767 df-map 6884 df-pm 6885 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-xneg 10105 df-xadd 10106 df-ico 10227 df-fz 10343 df-fzo 10477 df-seqfrec 10810 df-exp 10901 df-fac 11088 df-bc 11110 df-ihash 11139 df-shft 11500 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039 df-ef 12334 df-rest 13454 df-topgen 13473 df-psmet 14691 df-xmet 14692 df-met 14693 df-bl 14694 df-mopn 14695 df-top 14863 df-topon 14876 df-bases 14908 df-ntr 14961 df-cn 15053 df-cnp 15054 df-tx 15118 df-cncf 15436 df-limced 15521 df-dvap 15522

This theorem is referenced by: efle 15641 reefiso 15642 reapef 15643 logdivlti 15746 cxplt 15781 rpcxplt2 15784

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