reeff1olem - Intuitionistic Logic Explorer

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Theorem reeff1olem 15762

Description: Lemma for reeff1o 15764. (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 10311	. . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈)
2		0re 8290	. . . . 5 ⊢ 0 ∈ ℝ
3		iccssre 10307	. . . . 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 3253	. 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 8416	. . . . 5 ⊢ 0 < 1
10		1re 8289	. . . . . 6 ⊢ 1 ∈ ℝ
11		lttr 8363	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈))
12	2, 10, 11	mp3an12 1364	. . . . 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 8235	. . . 4 ⊢ ℝ ⊆ ℂ
16	5, 15	sstrdi 3254	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ)
17		efcn 15759	. . . 4 ⊢ exp ∈ (ℂ–cn→ℂ)
18	17	a1i 9	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ))
19		ssel2 3237	. . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ)
20	19	reefcld 12380	. . . 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 12383	. . . . 5 ⊢ (exp‘0) = 1
23		simpr 110	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈)
24	22, 23	eqbrtrid 4149	. . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈)
25		peano2re 8425	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ)
26	25	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ)
27		reefcl 12379	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ)
28	27	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ)
29		ltp1 9135	. . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1))
30	29	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1))
31	8	recnd 8318	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ)
32		ax-1cn 8236	. . . . . . 7 ⊢ 1 ∈ ℂ
33		addcom 8426	. . . . . . 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 10044	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ⁺)
36		efgt1p 12407	. . . . . . 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 4136	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈))
39	8, 26, 28, 30, 38	lttrd 8415	. . . 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 535	. . . . . . 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 529	. . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ (0[,]𝑈))
44	42, 43	sseldd 3243	. . . . 5 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 ∈ ℝ)
45		simprl 531	. . . . . 6 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑧 ∈ (0[,]𝑈))
46	42, 45	sseldd 3243	. . . . 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 533	. . . 4 ⊢ ((((𝑈 ∈ ℝ ∧ 1 < 𝑈) ∧ 𝑦 ∈ (0[,]𝑈)) ∧ (𝑧 ∈ (0[,]𝑈) ∧ 𝑦 < 𝑧)) → 𝑦 < 𝑧)
49		efltim 12409	. . . 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 15634	. 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈)
52		ssrexv 3307	. 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 1398 ∈ wcel 2205 ∃wrex 2523 ⊆ wss 3214 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 ℝ⁺crp 10004 (,)cioo 10240 [,]cicc 10243 expce 12353 –cn→ccncf 15561

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-pre-suploc 8264 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-ico 10246 df-icc 10247 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-rest 13538 df-topgen 13557 df-psmet 14817 df-xmet 14818 df-met 14819 df-bl 14820 df-mopn 14821 df-top 14989 df-topon 15002 df-bases 15034 df-ntr 15087 df-cn 15179 df-cnp 15180 df-tx 15244 df-cncf 15562 df-limced 15647 df-dvap 15648

This theorem is referenced by: reeff1oleme 15763 reeff1o 15764

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