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Mirrors > Home > MPE Home > Th. List > reeff1olem | Structured version Visualization version GIF version |
Description: Lemma for reeff1o 25752. (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 13304 | . . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈) | |
2 | 0re 11115 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | iccssre 13300 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (0[,]𝑈) ⊆ ℝ) | |
4 | 2, 3 | mpan 688 | . . . 4 ⊢ (𝑈 ∈ ℝ → (0[,]𝑈) ⊆ ℝ) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℝ) |
6 | 1, 5 | sstrid 3953 | . 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0(,)𝑈) ⊆ ℝ) |
7 | 2 | a1i 11 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 ∈ ℝ) |
8 | simpl 483 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ) | |
9 | 0lt1 11635 | . . . . 5 ⊢ 0 < 1 | |
10 | 1re 11113 | . . . . . 6 ⊢ 1 ∈ ℝ | |
11 | lttr 11189 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈)) | |
12 | 2, 10, 11 | mp3an12 1451 | . . . . 5 ⊢ (𝑈 ∈ ℝ → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈)) |
13 | 9, 12 | mpani 694 | . . . 4 ⊢ (𝑈 ∈ ℝ → (1 < 𝑈 → 0 < 𝑈)) |
14 | 13 | imp 407 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 < 𝑈) |
15 | ax-resscn 11066 | . . . 4 ⊢ ℝ ⊆ ℂ | |
16 | 5, 15 | sstrdi 3954 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ) |
17 | efcn 25748 | . . . 4 ⊢ exp ∈ (ℂ–cn→ℂ) | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ)) |
19 | ssel2 3937 | . . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ) | |
20 | 19 | reefcld 15924 | . . . 4 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ) |
21 | 5, 20 | sylan 580 | . . 3 ⊢ (((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ) |
22 | ef0 15927 | . . . . 5 ⊢ (exp‘0) = 1 | |
23 | simpr 485 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈) | |
24 | 22, 23 | eqbrtrid 5138 | . . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈) |
25 | peano2re 11286 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ) | |
26 | 25 | adantr 481 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ) |
27 | reefcl 15923 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ) | |
28 | 27 | adantr 481 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ) |
29 | ltp1 11953 | . . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1)) | |
30 | 29 | adantr 481 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1)) |
31 | 8 | recnd 11141 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ) |
32 | ax-1cn 11067 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
33 | addcom 11299 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑈 + 1) = (1 + 𝑈)) | |
34 | 31, 32, 33 | sylancl 586 | . . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) = (1 + 𝑈)) |
35 | 8, 14 | elrpd 12908 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ+) |
36 | efgt1p 15951 | . . . . . . 7 ⊢ (𝑈 ∈ ℝ+ → (1 + 𝑈) < (exp‘𝑈)) | |
37 | 35, 36 | syl 17 | . . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (1 + 𝑈) < (exp‘𝑈)) |
38 | 34, 37 | eqbrtrd 5125 | . . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈)) |
39 | 8, 26, 28, 30, 38 | lttrd 11274 | . . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (exp‘𝑈)) |
40 | 24, 39 | jca 512 | . . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ((exp‘0) < 𝑈 ∧ 𝑈 < (exp‘𝑈))) |
41 | 7, 8, 8, 14, 16, 18, 21, 40 | ivth 24764 | . 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈) |
42 | ssrexv 4009 | . 2 ⊢ ((0(,)𝑈) ⊆ ℝ → (∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)) | |
43 | 6, 41, 42 | sylc 65 | 1 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ⊆ wss 3908 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 < clt 11147 ℝ+crp 12869 (,)cioo 13218 [,]cicc 13221 expce 15898 –cn→ccncf 24185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-fac 14128 df-bc 14157 df-hash 14185 df-shft 14906 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-limsup 15307 df-clim 15324 df-rlim 15325 df-sum 15525 df-ef 15904 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-sca 17103 df-vsca 17104 df-ip 17105 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-hom 17111 df-cco 17112 df-rest 17258 df-topn 17259 df-0g 17277 df-gsum 17278 df-topgen 17279 df-pt 17280 df-prds 17283 df-xrs 17338 df-qtop 17343 df-imas 17344 df-xps 17346 df-mre 17420 df-mrc 17421 df-acs 17423 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-submnd 18556 df-mulg 18826 df-cntz 19050 df-cmn 19517 df-psmet 20735 df-xmet 20736 df-met 20737 df-bl 20738 df-mopn 20739 df-fbas 20740 df-fg 20741 df-cnfld 20744 df-top 22189 df-topon 22206 df-topsp 22228 df-bases 22242 df-cld 22316 df-ntr 22317 df-cls 22318 df-nei 22395 df-lp 22433 df-perf 22434 df-cn 22524 df-cnp 22525 df-haus 22612 df-tx 22859 df-hmeo 23052 df-fil 23143 df-fm 23235 df-flim 23236 df-flf 23237 df-xms 23619 df-ms 23620 df-tms 23621 df-cncf 24187 df-limc 25176 df-dv 25177 |
This theorem is referenced by: reeff1o 25752 |
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