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Mirrors > Home > ILE Home > Th. List > efltim | GIF version |
Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.) |
Ref | Expression |
---|---|
efltim | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 950 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) | |
2 | simp1 949 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) | |
3 | 1, 2 | resubcld 8010 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ) |
4 | posdif 8084 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | |
5 | 4 | biimp3a 1291 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 0 < (𝐵 − 𝐴)) |
6 | 3, 5 | elrpd 9328 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ+) |
7 | efgt1 11201 | . . . . 5 ⊢ ((𝐵 − 𝐴) ∈ ℝ+ → 1 < (exp‘(𝐵 − 𝐴))) | |
8 | 6, 7 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 < (exp‘(𝐵 − 𝐴))) |
9 | 2 | reefcld 11173 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘𝐴) ∈ ℝ) |
10 | 3 | reefcld 11173 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘(𝐵 − 𝐴)) ∈ ℝ) |
11 | efgt0 11188 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 0 < (exp‘𝐴)) | |
12 | 2, 11 | syl 14 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 0 < (exp‘𝐴)) |
13 | ltmulgt11 8480 | . . . . 5 ⊢ (((exp‘𝐴) ∈ ℝ ∧ (exp‘(𝐵 − 𝐴)) ∈ ℝ ∧ 0 < (exp‘𝐴)) → (1 < (exp‘(𝐵 − 𝐴)) ↔ (exp‘𝐴) < ((exp‘𝐴) · (exp‘(𝐵 − 𝐴))))) | |
14 | 9, 10, 12, 13 | syl3anc 1184 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (1 < (exp‘(𝐵 − 𝐴)) ↔ (exp‘𝐴) < ((exp‘𝐴) · (exp‘(𝐵 − 𝐴))))) |
15 | 8, 14 | mpbid 146 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘𝐴) < ((exp‘𝐴) · (exp‘(𝐵 − 𝐴)))) |
16 | 2 | recnd 7666 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ) |
17 | 3 | recnd 7666 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℂ) |
18 | efadd 11179 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (exp‘(𝐴 + (𝐵 − 𝐴))) = ((exp‘𝐴) · (exp‘(𝐵 − 𝐴)))) | |
19 | 16, 17, 18 | syl2anc 406 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘(𝐴 + (𝐵 − 𝐴))) = ((exp‘𝐴) · (exp‘(𝐵 − 𝐴)))) |
20 | 1 | recnd 7666 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ) |
21 | 16, 20 | pncan3d 7947 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
22 | 21 | fveq2d 5357 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘(𝐴 + (𝐵 − 𝐴))) = (exp‘𝐵)) |
23 | 19, 22 | eqtr3d 2134 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((exp‘𝐴) · (exp‘(𝐵 − 𝐴))) = (exp‘𝐵)) |
24 | 15, 23 | breqtrd 3899 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (exp‘𝐴) < (exp‘𝐵)) |
25 | 24 | 3expia 1151 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 930 = wceq 1299 ∈ wcel 1448 class class class wbr 3875 ‘cfv 5059 (class class class)co 5706 ℂcc 7498 ℝcr 7499 0cc0 7500 1c1 7501 + caddc 7503 · cmul 7505 < clt 7672 − cmin 7804 ℝ+crp 9291 expce 11146 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-disj 3853 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-frec 6218 df-1o 6243 df-oadd 6247 df-er 6359 df-en 6565 df-dom 6566 df-fin 6567 df-sup 6786 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-ico 9518 df-fz 9632 df-fzo 9761 df-seqfrec 10060 df-exp 10134 df-fac 10313 df-bc 10335 df-ihash 10363 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-clim 10887 df-sumdc 10962 df-ef 11152 |
This theorem is referenced by: efler 11203 reef11 11204 |
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