reeff1olem - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > reeff1olem		GIF version

Theorem reeff1olem 15214

Description: Lemma for reeff1o 15216. (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 10080	. . 3 ⊢ (0(,)𝑈) ⊆ (0[,]𝑈)
2		0re 8071	. . . . 5 ⊢ 0 ∈ ℝ
3		iccssre 10076	. . . . 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 3203	. 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 8198	. . . . 5 ⊢ 0 < 1
10		1re 8070	. . . . . 6 ⊢ 1 ∈ ℝ
11		lttr 8145	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑈) → 0 < 𝑈))
12	2, 10, 11	mp3an12 1339	. . . . 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 8016	. . . 4 ⊢ ℝ ⊆ ℂ
16	5, 15	sstrdi 3204	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (0[,]𝑈) ⊆ ℂ)
17		efcn 15211	. . . 4 ⊢ exp ∈ (ℂ–cn→ℂ)
18	17	a1i 9	. . 3 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → exp ∈ (ℂ–cn→ℂ))
19		ssel2 3187	. . . . 5 ⊢ (((0[,]𝑈) ⊆ ℝ ∧ 𝑦 ∈ (0[,]𝑈)) → 𝑦 ∈ ℝ)
20	19	reefcld 11951	. . . 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 11954	. . . . 5 ⊢ (exp‘0) = 1
23		simpr 110	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 1 < 𝑈)
24	22, 23	eqbrtrid 4078	. . . 4 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘0) < 𝑈)
25		peano2re 8207	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈ ℝ)
26	25	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) ∈ ℝ)
27		reefcl 11950	. . . . . 6 ⊢ (𝑈 ∈ ℝ → (exp‘𝑈) ∈ ℝ)
28	27	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (exp‘𝑈) ∈ ℝ)
29		ltp1 8916	. . . . . 6 ⊢ (𝑈 ∈ ℝ → 𝑈 < (𝑈 + 1))
30	29	adantr 276	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 < (𝑈 + 1))
31	8	recnd 8100	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℂ)
32		ax-1cn 8017	. . . . . . 7 ⊢ 1 ∈ ℂ
33		addcom 8208	. . . . . . 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 9814	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → 𝑈 ∈ ℝ⁺)
36		efgt1p 11978	. . . . . . 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 4065	. . . . 5 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → (𝑈 + 1) < (exp‘𝑈))
39	8, 26, 28, 30, 38	lttrd 8197	. . . 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 3193	. . . . 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 3193	. . . . 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 11980	. . . 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 15086	. 2 ⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ (0(,)𝑈)(exp‘𝑥) = 𝑈)
52		ssrexv 3257	. 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 1372 ∈ wcel 2175 ∃wrex 2484 ⊆ wss 3165 class class class wbr 4043 ‘cfv 5270 (class class class)co 5943 ℂcc 7922 ℝcr 7923 0cc0 7924 1c1 7925 + caddc 7927 < clt 8106 ℝ⁺crp 9774 (,)cioo 10009 [,]cicc 10012 expce 11924 –cn→ccncf 15013

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044 ax-pre-suploc 8045 ax-addf 8046 ax-mulf 8047

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-disj 4021 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-of 6157 df-1st 6225 df-2nd 6226 df-recs 6390 df-irdg 6455 df-frec 6476 df-1o 6501 df-oadd 6505 df-er 6619 df-map 6736 df-pm 6737 df-en 6827 df-dom 6828 df-fin 6829 df-sup 7085 df-inf 7086 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-xneg 9893 df-xadd 9894 df-ioo 10013 df-ico 10015 df-icc 10016 df-fz 10130 df-fzo 10264 df-seqfrec 10591 df-exp 10682 df-fac 10869 df-bc 10891 df-ihash 10919 df-shft 11097 df-cj 11124 df-re 11125 df-im 11126 df-rsqrt 11280 df-abs 11281 df-clim 11561 df-sumdc 11636 df-ef 11930 df-rest 13044 df-topgen 13063 df-psmet 14276 df-xmet 14277 df-met 14278 df-bl 14279 df-mopn 14280 df-top 14441 df-topon 14454 df-bases 14486 df-ntr 14539 df-cn 14631 df-cnp 14632 df-tx 14696 df-cncf 15014 df-limced 15099 df-dvap 15100

This theorem is referenced by: reeff1oleme 15215 reeff1o 15216

W3C validator