reeff1o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > reeff1o		Structured version Visualization version GIF version

Theorem reeff1o 26385

Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

**Assertion**
Ref	Expression
reeff1o	⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺

Proof of Theorem reeff1o Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeff1 16029	. 2 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ⁺
2		f1f 6719	. . . 4 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
3		ffn 6651	. . . 4 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → (exp ↾ ℝ) Fn ℝ)
4	1, 2, 3	mp2b 10	. . 3 ⊢ (exp ↾ ℝ) Fn ℝ
5		frn 6658	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → ran (exp ↾ ℝ) ⊆ ℝ⁺)
6	1, 2, 5	mp2b 10	. . . 4 ⊢ ran (exp ↾ ℝ) ⊆ ℝ⁺
7		elrp 12892	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧))
8		reclt1 12017	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ 0 < 𝑧) → (𝑧 < 1 ↔ 1 < (1 / 𝑧)))
9	7, 8	sylbi 217	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 ↔ 1 < (1 / 𝑧)))
10		rpre 12899	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
11		rpne0 12907	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ≠ 0)
12	10, 11	rereccld 11948	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → (1 / 𝑧) ∈ ℝ)
13		reeff1olem 26384	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑧) ∈ ℝ ∧ 1 < (1 / 𝑧)) → ∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧))
14	12, 13	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → ∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧))
15		eqcom 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 𝑧) = (exp‘𝑦) ↔ (exp‘𝑦) = (1 / 𝑧))
16		rpcnne0 12909	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
17		recn 11096	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
18		efcl 15989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (exp‘𝑦) ∈ ℂ)
20		efne0 16005	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ≠ 0)
21	17, 20	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (exp‘𝑦) ≠ 0)
22	19, 21	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0))
23		rec11r 11820	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) ∧ ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0)) → ((1 / 𝑧) = (exp‘𝑦) ↔ (1 / (exp‘𝑦)) = 𝑧))
24	16, 22, 23	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / 𝑧) = (exp‘𝑦) ↔ (1 / (exp‘𝑦)) = 𝑧))
25		efcan 16003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → ((exp‘𝑦) · (exp‘-𝑦)) = 1)
26	25	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℂ → 1 = ((exp‘𝑦) · (exp‘-𝑦)))
27		negcl 11360	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℂ → -𝑦 ∈ ℂ)
28		efcl 15989	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
30		ax-1cn 11064	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
31		divmul2 11780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℂ ∧ (exp‘-𝑦) ∈ ℂ ∧ ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0)) → ((1 / (exp‘𝑦)) = (exp‘-𝑦) ↔ 1 = ((exp‘𝑦) · (exp‘-𝑦))))
32	30, 31	mp3an1 1450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((exp‘-𝑦) ∈ ℂ ∧ ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0)) → ((1 / (exp‘𝑦)) = (exp‘-𝑦) ↔ 1 = ((exp‘𝑦) · (exp‘-𝑦))))
33	29, 18, 20, 32	syl12anc 836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℂ → ((1 / (exp‘𝑦)) = (exp‘-𝑦) ↔ 1 = ((exp‘𝑦) · (exp‘-𝑦))))
34	26, 33	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (1 / (exp‘𝑦)) = (exp‘-𝑦))
35	17, 34	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (1 / (exp‘𝑦)) = (exp‘-𝑦))
36	35	eqeq1d 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → ((1 / (exp‘𝑦)) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
37	36	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / (exp‘𝑦)) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
38	24, 37	bitrd 279	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / 𝑧) = (exp‘𝑦) ↔ (exp‘-𝑦) = 𝑧))
39	15, 38	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((exp‘𝑦) = (1 / 𝑧) ↔ (exp‘-𝑦) = 𝑧))
40	39	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((exp‘𝑦) = (1 / 𝑧) → (exp‘-𝑦) = 𝑧))
41	40	reximdva 3145	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → (∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧) → ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧))
42	41	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → (∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧) → ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧))
43	14, 42	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧)
44		renegcl 11424	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
45		infm3lem 12080	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)
46		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = -𝑦 → ((exp‘𝑥) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
47	44, 45, 46	rexxfr 5354	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧 ↔ ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧)
48	43, 47	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
49	48	ex 412	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → (1 < (1 / 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧))
50	9, 49	sylbid 240	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧))
51	50	imp 406	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑧 < 1) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
52		ef0 15998	. . . . . . . . . . 11 ⊢ (exp‘0) = 1
53	52	eqeq2i 2744	. . . . . . . . . 10 ⊢ (𝑧 = (exp‘0) ↔ 𝑧 = 1)
54		0re 11114	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
55		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((exp‘𝑥) = 𝑧 ↔ (exp‘0) = 𝑧))
56	55	rspcev 3577	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ (exp‘0) = 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
57	54, 56	mpan 690	. . . . . . . . . . 11 ⊢ ((exp‘0) = 𝑧 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
58	57	eqcoms 2739	. . . . . . . . . 10 ⊢ (𝑧 = (exp‘0) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
59	53, 58	sylbir 235	. . . . . . . . 9 ⊢ (𝑧 = 1 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
60	59	adantl 481	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑧 = 1) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
61		reeff1olem 26384	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
62	10, 61	sylan 580	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
63		1re 11112	. . . . . . . . 9 ⊢ 1 ∈ ℝ
64		lttri4 11197	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧))
65	10, 63, 64	sylancl 586	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧))
66	51, 60, 62, 65	mpjao3dan 1434	. . . . . . 7 ⊢ (𝑧 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
67		fvres 6841	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
68	67	eqeq1d 2733	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ (exp‘𝑥) = 𝑧))
69	68	rexbiia 3077	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
70	66, 69	sylibr 234	. . . . . 6 ⊢ (𝑧 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧)
71		fvelrnb 6882	. . . . . . 7 ⊢ ((exp ↾ ℝ) Fn ℝ → (𝑧 ∈ ran (exp ↾ ℝ) ↔ ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧))
72	4, 71	ax-mp 5	. . . . . 6 ⊢ (𝑧 ∈ ran (exp ↾ ℝ) ↔ ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧)
73	70, 72	sylibr 234	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ran (exp ↾ ℝ))
74	73	ssriv 3938	. . . 4 ⊢ ℝ⁺ ⊆ ran (exp ↾ ℝ)
75	6, 74	eqssi 3951	. . 3 ⊢ ran (exp ↾ ℝ) = ℝ⁺
76		df-fo 6487	. . 3 ⊢ ((exp ↾ ℝ):ℝ–onto→ℝ⁺ ↔ ((exp ↾ ℝ) Fn ℝ ∧ ran (exp ↾ ℝ) = ℝ⁺))
77	4, 75, 76	mpbir2an 711	. 2 ⊢ (exp ↾ ℝ):ℝ–onto→ℝ⁺
78		df-f1o 6488	. 2 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ↔ ((exp ↾ ℝ):ℝ–1-1→ℝ⁺ ∧ (exp ↾ ℝ):ℝ–onto→ℝ⁺))
79	1, 77, 78	mpbir2an 711	1 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 ⊆ wss 3902 class class class wbr 5091 ran crn 5617 ↾ cres 5618 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 · cmul 11011 < clt 11146 -cneg 11345 / cdiv 11774 ℝ⁺crp 12890 expce 15968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19230 df-cmn 19695 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-fbas 21289 df-fg 21290 df-cnfld 21293 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796

This theorem is referenced by: reefiso 26386 efcvx 26387 reefgim 26388 eff1olem 26485 dfrelog 26502 relogf1o 26503 dvrelog 26574

W3C validator