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Theorem eflt 14235

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 11708	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
2		efcn 14228	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
3		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℝ)
4	3	recnd 7988	. . . . 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 11679	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) ∈ ℝ)
8	3	reefcld 11679	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐵) ∈ ℝ)
9		difrp 9694	. . . . . . 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 14104	. . . . 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 1342	. . . 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 5900	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 − 𝐵)) = (abs‘(𝐴 − 𝐵)))
19	18	breq1d 4015	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((abs‘(𝑥 − 𝐵)) < 𝑑 ↔ (abs‘(𝐴 − 𝐵)) < 𝑑))
20	19	imbrov2fvoveq 5902	. . . . . 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 7988	. . . . . 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 2848	. . . . 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 14234	. . . 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 2599	. . 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 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 ℝcr 7812 < clt 7994 − cmin 8130 ℝ⁺crp 9655 abscabs 11008 expce 11652 –cn→ccncf 14096

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-addf 7935 ax-mulf 7936

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-map 6652 df-pm 6653 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-ico 9896 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-shft 10826 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-rest 12695 df-topgen 12714 df-psmet 13486 df-xmet 13487 df-met 13488 df-bl 13489 df-mopn 13490 df-top 13537 df-topon 13550 df-bases 13582 df-ntr 13635 df-cn 13727 df-cnp 13728 df-tx 13792 df-cncf 14097 df-limced 14164 df-dvap 14165

This theorem is referenced by: efle 14236 reefiso 14237 reapef 14238 logdivlti 14341 cxplt 14375 rpcxplt2 14378

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