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Mirrors > Home > ILE Home > Th. List > reeff1olem | GIF version |
Description: Lemma for reeff1o 13033. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
reeff1olem | ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossicc 9841 | . . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈) | |
2 | 0re 7857 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | iccssre 9837 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (0[,]𝑈) ⊆ ℝ) | |
4 | 2, 3 | mpan 421 | . . . 4 ⊢ (𝑈 ∈ ℝ → (0[,]𝑈) ⊆ ℝ) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℝ) |
6 | 1, 5 | sstrid 3135 | . 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0(,)𝑈) ⊆ ℝ) |
7 | 2 | a1i 9 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 ∈ ℝ) |
8 | simpl 108 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ) | |
9 | 0lt1 7981 | . . . . 5 ⊢ 0 < 1 | |
10 | 1re 7856 | . . . . . 6 ⊢ 1 ∈ ℝ | |
11 | lttr 7930 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈)) | |
12 | 2, 10, 11 | mp3an12 1306 | . . . . 5 ⊢ (𝑈 ∈ ℝ → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈)) |
13 | 9, 12 | mpani 427 | . . . 4 ⊢ (𝑈 ∈ ℝ → (1 < 𝑈 → 0 < 𝑈)) |
14 | 13 | imp 123 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 < 𝑈) |
15 | ax-resscn 7803 | . . . 4 ⊢ ℝ ⊆ ℂ | |
16 | 5, 15 | sstrdi 3136 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ) |
17 | efcn 13028 | . . . 4 ⊢ exp ∈ (ℂ–cn→ℂ) | |
18 | 17 | a1i 9 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ)) |
19 | ssel2 3119 | . . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ) | |
20 | 19 | reefcld 11543 | . . . 4 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ) |
21 | 5, 20 | sylan 281 | . . 3 ⊢ (((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ) |
22 | ef0 11546 | . . . . 5 ⊢ (exp‘0) = 1 | |
23 | simpr 109 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈) | |
24 | 22, 23 | eqbrtrid 3995 | . . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈) |
25 | peano2re 7990 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ) | |
26 | 25 | adantr 274 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ) |
27 | reefcl 11542 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ) | |
28 | 27 | adantr 274 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ) |
29 | ltp1 8694 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1)) | |
30 | 29 | adantr 274 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1)) |
31 | 8 | recnd 7885 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ) |
32 | ax-1cn 7804 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
33 | addcom 7991 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑈 + 1) = (1 + 𝑈)) | |
34 | 31, 32, 33 | sylancl 410 | . . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) = (1 + 𝑈)) |
35 | 8, 14 | elrpd 9578 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ+) |
36 | efgt1p 11570 | . . . . . . 7 ⊢ (𝑈 ∈ ℝ+ → (1 + 𝑈) < (exp‘𝑈)) | |
37 | 35, 36 | syl 14 | . . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (1 + 𝑈) < (exp‘𝑈)) |
38 | 34, 37 | eqbrtrd 3982 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈)) |
39 | 8, 26, 28, 30, 38 | lttrd 7980 | . . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (exp‘𝑈)) |
40 | 24, 39 | jca 304 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ((exp‘0) < 𝑈 ∧ 𝑈 < (exp‘𝑈))) |
41 | simplll 523 | . . . . . . 7 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑈 ∈ ℝ) | |
42 | 2, 41, 3 | sylancr 411 | . . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → (0[,]𝑈) ⊆ ℝ) |
43 | simplr 520 | . . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ (0[,]𝑈)) | |
44 | 42, 43 | sseldd 3125 | . . . . 5 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ ℝ) |
45 | simprl 521 | . . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑧 ∈ (0[,]𝑈)) | |
46 | 42, 45 | sseldd 3125 | . . . . 5 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑧 ∈ ℝ) |
47 | 44, 46 | jca 304 | . . . 4 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) |
48 | simprr 522 | . . . 4 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 < 𝑧) | |
49 | efltim 11572 | . . . 4 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧 → (exp‘𝑦) < (exp‘𝑧))) | |
50 | 47, 48, 49 | sylc 62 | . . 3 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → (exp‘𝑦) < (exp‘𝑧)) |
51 | 7, 8, 8, 14, 16, 18, 21, 40, 50 | ivthinc 12960 | . 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈) |
52 | ssrexv 3189 | . 2 ⊢ ((0(,)𝑈) ⊆ ℝ → (∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)) | |
53 | 6, 51, 52 | sylc 62 | 1 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 2125 ∃wrex 2433 ⊆ wss 3098 class class class wbr 3961 ‘cfv 5163 (class class class)co 5814 ℂcc 7709 ℝcr 7710 0cc0 7711 1c1 7712 + caddc 7714 < clt 7891 ℝ+crp 9538 (,)cioo 9770 [,]cicc 9773 expce 11516 –cn→ccncf 12896 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 ax-pre-suploc 7832 ax-addf 7833 ax-mulf 7834 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-disj 3939 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-of 6022 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-oadd 6357 df-er 6469 df-map 6584 df-pm 6585 df-en 6675 df-dom 6676 df-fin 6677 df-sup 6916 df-inf 6917 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-xneg 9657 df-xadd 9658 df-ioo 9774 df-ico 9776 df-icc 9777 df-fz 9891 df-fzo 10020 df-seqfrec 10323 df-exp 10397 df-fac 10577 df-bc 10599 df-ihash 10627 df-shft 10692 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-sumdc 11228 df-ef 11522 df-rest 12292 df-topgen 12311 df-psmet 12326 df-xmet 12327 df-met 12328 df-bl 12329 df-mopn 12330 df-top 12335 df-topon 12348 df-bases 12380 df-ntr 12435 df-cn 12527 df-cnp 12528 df-tx 12592 df-cncf 12897 df-limced 12964 df-dvap 12965 |
This theorem is referenced by: reeff1oleme 13032 reeff1o 13033 |
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