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Theorem reeff1olem 14085

Description: Lemma for reeff1o 14087. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

**Assertion**
Ref	Expression
reeff1olem	⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)

Distinct variable group: 𝑥,𝑈

Proof of Theorem reeff1olem Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossicc 9957	. . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈)
2		0re 7956	. . . . 5 ⊢ 0 ∈ ℝ
3		iccssre 9953	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (0[,]𝑈) ⊆ ℝ)
4	2, 3	mpan 424	. . . 4 ⊢ (𝑈 ∈ ℝ → (0[,]𝑈) ⊆ ℝ)
5	4	adantr 276	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℝ)
6	1, 5	sstrid 3166	. 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0(,)𝑈) ⊆ ℝ)
7	2	a1i 9	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 ∈ ℝ)
8		simpl 109	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ)
9		0lt1 8082	. . . . 5 ⊢ 0 < 1
10		1re 7955	. . . . . 6 ⊢ 1 ∈ ℝ
11		lttr 8029	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈))
12	2, 10, 11	mp3an12 1327	. . . . 5 ⊢ (𝑈 ∈ ℝ → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈))
13	9, 12	mpani 430	. . . 4 ⊢ (𝑈 ∈ ℝ → (1 < 𝑈 → 0 < 𝑈))
14	13	imp 124	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 0 < 𝑈)
15		ax-resscn 7902	. . . 4 ⊢ ℝ ⊆ ℂ
16	5, 15	sstrdi 3167	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ)
17		efcn 14082	. . . 4 ⊢ exp ∈ (ℂ–cn→ℂ)
18	17	a1i 9	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ))
19		ssel2 3150	. . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ)
20	19	reefcld 11672	. . . 4 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ)
21	5, 20	sylan 283	. . 3 ⊢ (((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) → (exp‘𝑦) ∈ ℝ)
22		ef0 11675	. . . . 5 ⊢ (exp‘0) = 1
23		simpr 110	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈)
24	22, 23	eqbrtrid 4038	. . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈)
25		peano2re 8091	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ)
26	25	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ)
27		reefcl 11671	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ)
28	27	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ)
29		ltp1 8799	. . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1))
30	29	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1))
31	8	recnd 7984	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ)
32		ax-1cn 7903	. . . . . . 7 ⊢ 1 ∈ ℂ
33		addcom 8092	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑈 + 1) = (1 + 𝑈))
34	31, 32, 33	sylancl 413	. . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) = (1 + 𝑈))
35	8, 14	elrpd 9691	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ⁺)
36		efgt1p 11699	. . . . . . 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 4025	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈))
39	8, 26, 28, 30, 38	lttrd 8081	. . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (exp‘𝑈))
40	24, 39	jca 306	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ((exp‘0) < 𝑈 ∧ 𝑈 < (exp‘𝑈)))
41		simplll 533	. . . . . . 7 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑈 ∈ ℝ)
42	2, 41, 3	sylancr 414	. . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → (0[,]𝑈) ⊆ ℝ)
43		simplr 528	. . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ (0[,]𝑈))
44	42, 43	sseldd 3156	. . . . 5 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ ℝ)
45		simprl 529	. . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑧 ∈ (0[,]𝑈))
46	42, 45	sseldd 3156	. . . . 5 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑧 ∈ ℝ)
47	44, 46	jca 306	. . . 4 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
48		simprr 531	. . . 4 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 < 𝑧)
49		efltim 11701	. . . 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 14014	. 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈)
52		ssrexv 3220	. 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 104 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 + caddc 7813 < clt 7990 ℝ⁺crp 9651 (,)cioo 9886 [,]cicc 9889 expce 11645 –cn→ccncf 13950

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 ax-pre-suploc 7931 ax-addf 7932 ax-mulf 7933

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-disj 3981 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-of 6082 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-map 6649 df-pm 6650 df-en 6740 df-dom 6741 df-fin 6742 df-sup 6982 df-inf 6983 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-xneg 9770 df-xadd 9771 df-ioo 9890 df-ico 9892 df-icc 9893 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-fac 10701 df-bc 10723 df-ihash 10751 df-shft 10819 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-sumdc 11357 df-ef 11651 df-rest 12680 df-topgen 12699 df-psmet 13338 df-xmet 13339 df-met 13340 df-bl 13341 df-mopn 13342 df-top 13389 df-topon 13402 df-bases 13434 df-ntr 13489 df-cn 13581 df-cnp 13582 df-tx 13646 df-cncf 13951 df-limced 14018 df-dvap 14019

This theorem is referenced by: reeff1oleme 14086 reeff1o 14087

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