reeff1olem - Intuitionistic Logic Explorer

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Theorem reeff1olem 15565

Description: Lemma for reeff1o 15567. (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 10238	. . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈)
2		0re 8222	. . . . 5 ⊢ 0 ∈ ℝ
3		iccssre 10234	. . . . 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 3239	. 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 8348	. . . . 5 ⊢ 0 < 1
10		1re 8221	. . . . . 6 ⊢ 1 ∈ ℝ
11		lttr 8295	. . . . . 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 8167	. . . 4 ⊢ ℝ ⊆ ℂ
16	5, 15	sstrdi 3240	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ)
17		efcn 15562	. . . 4 ⊢ exp ∈ (ℂ–cn→ℂ)
18	17	a1i 9	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ))
19		ssel2 3223	. . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ)
20	19	reefcld 12293	. . . 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 12296	. . . . 5 ⊢ (exp‘0) = 1
23		simpr 110	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈)
24	22, 23	eqbrtrid 4128	. . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈)
25		peano2re 8357	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ)
26	25	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ)
27		reefcl 12292	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ)
28	27	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ)
29		ltp1 9066	. . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1))
30	29	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1))
31	8	recnd 8250	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ)
32		ax-1cn 8168	. . . . . . 7 ⊢ 1 ∈ ℂ
33		addcom 8358	. . . . . . 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 9972	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ⁺)
36		efgt1p 12320	. . . . . . 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 4115	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈))
39	8, 26, 28, 30, 38	lttrd 8347	. . . 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 3229	. . . . 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 3229	. . . . 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 12322	. . . 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 15437	. 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈)
52		ssrexv 3293	. 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 2202 ∃wrex 2512 ⊆ wss 3201 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 ℝcr 8074 0cc0 8075 1c1 8076 + caddc 8078 < clt 8256 ℝ⁺crp 9932 (,)cioo 10167 [,]cicc 10170 expce 12266 –cn→ccncf 15364

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 ax-pre-suploc 8196 ax-addf 8197 ax-mulf 8198

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-disj 4070 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-oadd 6629 df-er 6745 df-map 6862 df-pm 6863 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-xneg 10051 df-xadd 10052 df-ioo 10171 df-ico 10173 df-icc 10174 df-fz 10289 df-fzo 10423 df-seqfrec 10756 df-exp 10847 df-fac 11034 df-bc 11056 df-ihash 11084 df-shft 11438 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-clim 11902 df-sumdc 11977 df-ef 12272 df-rest 13387 df-topgen 13406 df-psmet 14622 df-xmet 14623 df-met 14624 df-bl 14625 df-mopn 14626 df-top 14792 df-topon 14805 df-bases 14837 df-ntr 14890 df-cn 14982 df-cnp 14983 df-tx 15047 df-cncf 15365 df-limced 15450 df-dvap 15451

This theorem is referenced by: reeff1oleme 15566 reeff1o 15567

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