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Theorem eflt 15766

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 12409	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
2		efcn 15759	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
3		simplr 529	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℝ)
4	3	recnd 8318	. . . . 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 12380	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) ∈ ℝ)
8	3	reefcld 12380	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐵) ∈ ℝ)
9		difrp 10043	. . . . . . 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 15569	. . . . 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 6081	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 − 𝐵)) = (abs‘(𝐴 − 𝐵)))
19	18	breq1d 4124	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((abs‘(𝑥 − 𝐵)) < 𝑑 ↔ (abs‘(𝐴 − 𝐵)) < 𝑑))
20	19	imbrov2fvoveq 6083	. . . . . 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 8318	. . . . . 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 2928	. . . . 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 15765	. . . 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 2667	. . 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 2205 ∀wral 2522 ∃wrex 2523 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 < clt 8324 − cmin 8460 ℝ⁺crp 10004 abscabs 11707 expce 12353 –cn→ccncf 15561

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-addf 8265 ax-mulf 8266

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ico 10246 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-rest 13538 df-topgen 13557 df-psmet 14817 df-xmet 14818 df-met 14819 df-bl 14820 df-mopn 14821 df-top 14989 df-topon 15002 df-bases 15034 df-ntr 15087 df-cn 15179 df-cnp 15180 df-tx 15244 df-cncf 15562 df-limced 15647 df-dvap 15648

This theorem is referenced by: efle 15767 reefiso 15768 reapef 15769 logdivlti 15872 cxplt 15907 rpcxplt2 15910

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