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Theorem eflt 15317

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 12079	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
2		efcn 15310	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
3		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → 𝐵 ∈ ℝ)
4	3	recnd 8116	. . . . 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 12050	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐴) ∈ ℝ)
8	3	reefcld 12050	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (exp‘𝐴) < (exp‘𝐵)) → (exp‘𝐵) ∈ ℝ)
9		difrp 9829	. . . . . . 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 15120	. . . . 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 1355	. . . 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 5979	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 − 𝐵)) = (abs‘(𝐴 − 𝐵)))
19	18	breq1d 4060	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((abs‘(𝑥 − 𝐵)) < 𝑑 ↔ (abs‘(𝐴 − 𝐵)) < 𝑑))
20	19	imbrov2fvoveq 5981	. . . . . 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 8116	. . . . . 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 2886	. . . . 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 15316	. . . 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 2629	. . 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 1373 ∈ wcel 2177 ∀wral 2485 ∃wrex 2486 class class class wbr 4050 ‘cfv 5279 (class class class)co 5956 ℂcc 7938 ℝcr 7939 < clt 8122 − cmin 8258 ℝ⁺crp 9790 abscabs 11378 expce 12023 –cn→ccncf 15112

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4166 ax-sep 4169 ax-nul 4177 ax-pow 4225 ax-pr 4260 ax-un 4487 ax-setind 4592 ax-iinf 4643 ax-cnex 8031 ax-resscn 8032 ax-1cn 8033 ax-1re 8034 ax-icn 8035 ax-addcl 8036 ax-addrcl 8037 ax-mulcl 8038 ax-mulrcl 8039 ax-addcom 8040 ax-mulcom 8041 ax-addass 8042 ax-mulass 8043 ax-distr 8044 ax-i2m1 8045 ax-0lt1 8046 ax-1rid 8047 ax-0id 8048 ax-rnegex 8049 ax-precex 8050 ax-cnre 8051 ax-pre-ltirr 8052 ax-pre-ltwlin 8053 ax-pre-lttrn 8054 ax-pre-apti 8055 ax-pre-ltadd 8056 ax-pre-mulgt0 8057 ax-pre-mulext 8058 ax-arch 8059 ax-caucvg 8060 ax-addf 8062 ax-mulf 8063

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3622 df-sn 3643 df-pr 3644 df-op 3646 df-uni 3856 df-int 3891 df-iun 3934 df-disj 4027 df-br 4051 df-opab 4113 df-mpt 4114 df-tr 4150 df-id 4347 df-po 4350 df-iso 4351 df-iord 4420 df-on 4422 df-ilim 4423 df-suc 4425 df-iom 4646 df-xp 4688 df-rel 4689 df-cnv 4690 df-co 4691 df-dm 4692 df-rn 4693 df-res 4694 df-ima 4695 df-iota 5240 df-fun 5281 df-fn 5282 df-f 5283 df-f1 5284 df-fo 5285 df-f1o 5286 df-fv 5287 df-isom 5288 df-riota 5911 df-ov 5959 df-oprab 5960 df-mpo 5961 df-of 6170 df-1st 6238 df-2nd 6239 df-recs 6403 df-irdg 6468 df-frec 6489 df-1o 6514 df-oadd 6518 df-er 6632 df-map 6749 df-pm 6750 df-en 6840 df-dom 6841 df-fin 6842 df-sup 7100 df-inf 7101 df-pnf 8124 df-mnf 8125 df-xr 8126 df-ltxr 8127 df-le 8128 df-sub 8260 df-neg 8261 df-reap 8663 df-ap 8670 df-div 8761 df-inn 9052 df-2 9110 df-3 9111 df-4 9112 df-n0 9311 df-z 9388 df-uz 9664 df-q 9756 df-rp 9791 df-xneg 9909 df-xadd 9910 df-ico 10031 df-fz 10146 df-fzo 10280 df-seqfrec 10610 df-exp 10701 df-fac 10888 df-bc 10910 df-ihash 10938 df-shft 11196 df-cj 11223 df-re 11224 df-im 11225 df-rsqrt 11379 df-abs 11380 df-clim 11660 df-sumdc 11735 df-ef 12029 df-rest 13143 df-topgen 13162 df-psmet 14375 df-xmet 14376 df-met 14377 df-bl 14378 df-mopn 14379 df-top 14540 df-topon 14553 df-bases 14585 df-ntr 14638 df-cn 14730 df-cnp 14731 df-tx 14795 df-cncf 15113 df-limced 15198 df-dvap 15199

This theorem is referenced by: efle 15318 reefiso 15319 reapef 15320 logdivlti 15423 cxplt 15458 rpcxplt2 15461

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