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Mirrors > Home > ILE Home > Th. List > reeff1o | GIF version |
Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.) |
Ref | Expression |
---|---|
reeff1o | ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reeff1 11627 | . 2 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ+ | |
2 | f1f 5387 | . . . 4 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
3 | ffn 5331 | . . . 4 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → (exp ↾ ℝ) Fn ℝ) | |
4 | 1, 2, 3 | mp2b 8 | . . 3 ⊢ (exp ↾ ℝ) Fn ℝ |
5 | frn 5340 | . . . . 5 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → ran (exp ↾ ℝ) ⊆ ℝ+) | |
6 | 1, 2, 5 | mp2b 8 | . . . 4 ⊢ ran (exp ↾ ℝ) ⊆ ℝ+ |
7 | rpre 9587 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ) | |
8 | reeff1olem 13233 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) | |
9 | 7, 8 | sylan 281 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℝ+ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) |
10 | 7 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ+ ∧ 𝑧 < e) → 𝑧 ∈ ℝ) |
11 | rpgt0 9592 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 0 < 𝑧) | |
12 | 11 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ+ ∧ 𝑧 < e) → 0 < 𝑧) |
13 | simpr 109 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ+ ∧ 𝑧 < e) → 𝑧 < e) | |
14 | 0xr 7936 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ* | |
15 | ere 11597 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ | |
16 | 15 | rexri 7947 | . . . . . . . . . . 11 ⊢ e ∈ ℝ* |
17 | elioo2 9848 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℝ* ∧ e ∈ ℝ*) → (𝑧 ∈ (0(,)e) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 < e))) | |
18 | 14, 16, 17 | mp2an 423 | . . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,)e) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 < e)) |
19 | reeff1oleme 13234 | . . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,)e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) | |
20 | 18, 19 | sylbir 134 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 < e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) |
21 | 10, 12, 13, 20 | syl3anc 1227 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℝ+ ∧ 𝑧 < e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) |
22 | 1lt2 9017 | . . . . . . . . . 10 ⊢ 1 < 2 | |
23 | egt2lt3 11706 | . . . . . . . . . . 11 ⊢ (2 < e ∧ e < 3) | |
24 | 23 | simpli 110 | . . . . . . . . . 10 ⊢ 2 < e |
25 | 1re 7889 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
26 | 2re 8918 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
27 | 25, 26, 15 | lttri 7994 | . . . . . . . . . 10 ⊢ ((1 < 2 ∧ 2 < e) → 1 < e) |
28 | 22, 24, 27 | mp2an 423 | . . . . . . . . 9 ⊢ 1 < e |
29 | 1red 7905 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 1 ∈ ℝ) | |
30 | 15 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → e ∈ ℝ) |
31 | axltwlin 7957 | . . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ e ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 < e → (1 < 𝑧 ∨ 𝑧 < e))) | |
32 | 29, 30, 7, 31 | syl3anc 1227 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → (1 < e → (1 < 𝑧 ∨ 𝑧 < e))) |
33 | 28, 32 | mpi 15 | . . . . . . . 8 ⊢ (𝑧 ∈ ℝ+ → (1 < 𝑧 ∨ 𝑧 < e)) |
34 | 9, 21, 33 | mpjaodan 788 | . . . . . . 7 ⊢ (𝑧 ∈ ℝ+ → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) |
35 | fvres 5504 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥)) | |
36 | 35 | eqeq1d 2173 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ (exp‘𝑥) = 𝑧)) |
37 | 36 | rexbiia 2479 | . . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧) |
38 | 34, 37 | sylibr 133 | . . . . . 6 ⊢ (𝑧 ∈ ℝ+ → ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧) |
39 | fvelrnb 5528 | . . . . . . 7 ⊢ ((exp ↾ ℝ) Fn ℝ → (𝑧 ∈ ran (exp ↾ ℝ) ↔ ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧)) | |
40 | 4, 39 | ax-mp 5 | . . . . . 6 ⊢ (𝑧 ∈ ran (exp ↾ ℝ) ↔ ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧) |
41 | 38, 40 | sylibr 133 | . . . . 5 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ran (exp ↾ ℝ)) |
42 | 41 | ssriv 3141 | . . . 4 ⊢ ℝ+ ⊆ ran (exp ↾ ℝ) |
43 | 6, 42 | eqssi 3153 | . . 3 ⊢ ran (exp ↾ ℝ) = ℝ+ |
44 | df-fo 5188 | . . 3 ⊢ ((exp ↾ ℝ):ℝ–onto→ℝ+ ↔ ((exp ↾ ℝ) Fn ℝ ∧ ran (exp ↾ ℝ) = ℝ+)) | |
45 | 4, 43, 44 | mpbir2an 931 | . 2 ⊢ (exp ↾ ℝ):ℝ–onto→ℝ+ |
46 | df-f1o 5189 | . 2 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ↔ ((exp ↾ ℝ):ℝ–1-1→ℝ+ ∧ (exp ↾ ℝ):ℝ–onto→ℝ+)) | |
47 | 1, 45, 46 | mpbir2an 931 | 1 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 ⊆ wss 3111 class class class wbr 3976 ran crn 4599 ↾ cres 4600 Fn wfn 5177 ⟶wf 5178 –1-1→wf1 5179 –onto→wfo 5180 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 1c1 7745 ℝ*cxr 7923 < clt 7924 2c2 8899 3c3 8900 ℝ+crp 9580 (,)cioo 9815 expce 11569 eceu 11570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 ax-pre-suploc 7865 ax-addf 7866 ax-mulf 7867 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-map 6607 df-pm 6608 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-ioo 9819 df-ico 9821 df-icc 9822 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-e 11576 df-rest 12494 df-topgen 12513 df-psmet 12528 df-xmet 12529 df-met 12530 df-bl 12531 df-mopn 12532 df-top 12537 df-topon 12550 df-bases 12582 df-ntr 12637 df-cn 12729 df-cnp 12730 df-tx 12794 df-cncf 13099 df-limced 13166 df-dvap 13167 |
This theorem is referenced by: reefiso 13239 dfrelog 13322 relogf1o 13323 reeflog 13325 |
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