reeff1o - Metamath Proof Explorer

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Theorem reeff1o 26434

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 16082	. 2 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ⁺
2		f1f 6727	. . . 4 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
3		ffn 6659	. . . 4 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → (exp ↾ ℝ) Fn ℝ)
4	1, 2, 3	mp2b 10	. . 3 ⊢ (exp ↾ ℝ) Fn ℝ
5		frn 6666	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → ran (exp ↾ ℝ) ⊆ ℝ⁺)
6	1, 2, 5	mp2b 10	. . . 4 ⊢ ran (exp ↾ ℝ) ⊆ ℝ⁺
7		elrp 12939	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧))
8		reclt1 12046	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ 0 < 𝑧) → (𝑧 < 1 ↔ 1 < (1 / 𝑧)))
9	7, 8	sylbi 219	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 ↔ 1 < (1 / 𝑧)))
10		rpre 12946	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
11		rpne0 12954	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ≠ 0)
12	10, 11	rereccld 11977	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → (1 / 𝑧) ∈ ℝ)
13		reeff1olem 26433	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑧) ∈ ℝ ∧ 1 < (1 / 𝑧)) → ∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧))
14	12, 13	sylan 587	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → ∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧))
15		eqcom 2748	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 𝑧) = (exp‘𝑦) ↔ (exp‘𝑦) = (1 / 𝑧))
16		rpcnne0 12956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
17		recn 11123	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
18		efcl 16042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (exp‘𝑦) ∈ ℂ)
20		efne0 16058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ≠ 0)
21	17, 20	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (exp‘𝑦) ≠ 0)
22	19, 21	jca 517	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0))
23		rec11r 11849	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) ∧ ((exp‘𝑦) ∈ ℂ ∧ (exp‘𝑦) ≠ 0)) → ((1 / 𝑧) = (exp‘𝑦) ↔ (1 / (exp‘𝑦)) = 𝑧))
24	16, 22, 23	syl2an 603	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / 𝑧) = (exp‘𝑦) ↔ (1 / (exp‘𝑦)) = 𝑧))
25		efcan 16056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → ((exp‘𝑦) · (exp‘-𝑦)) = 1)
26	25	eqcomd 2747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℂ → 1 = ((exp‘𝑦) · (exp‘-𝑦)))
27		negcl 11388	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℂ → -𝑦 ∈ ℂ)
28		efcl 16042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
30		ax-1cn 11091	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
31		divmul2 11808	. . . . . . . . . . . . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . . . . . . . . . . . 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 843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℂ → ((1 / (exp‘𝑦)) = (exp‘-𝑦) ↔ 1 = ((exp‘𝑦) · (exp‘-𝑦))))
34	26, 33	mpbird 259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (1 / (exp‘𝑦)) = (exp‘-𝑦))
35	17, 34	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (1 / (exp‘𝑦)) = (exp‘-𝑦))
36	35	eqeq1d 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → ((1 / (exp‘𝑦)) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
37	36	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / (exp‘𝑦)) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
38	24, 37	bitrd 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((1 / 𝑧) = (exp‘𝑦) ↔ (exp‘-𝑦) = 𝑧))
39	15, 38	bitr3id 287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((exp‘𝑦) = (1 / 𝑧) ↔ (exp‘-𝑦) = 𝑧))
40	39	biimpd 231	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → ((exp‘𝑦) = (1 / 𝑧) → (exp‘-𝑦) = 𝑧))
41	40	reximdva 3154	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → (∃𝑦 ∈ ℝ (exp‘𝑦) = (1 / 𝑧) → ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧))
42	41	adantr 482	. . . . . . . . . . . . 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 11452	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
45		infm3lem 12109	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)
46		fveqeq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = -𝑦 → ((exp‘𝑥) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
47	44, 45, 46	rexxfr 5348	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧 ↔ ∃𝑦 ∈ ℝ (exp‘-𝑦) = 𝑧)
48	43, 47	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < (1 / 𝑧)) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
49	48	ex 414	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → (1 < (1 / 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧))
50	9, 49	sylbid 242	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧))
51	50	imp 408	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑧 < 1) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
52		ef0 16051	. . . . . . . . . . 11 ⊢ (exp‘0) = 1
53	52	eqeq2i 2754	. . . . . . . . . 10 ⊢ (𝑧 = (exp‘0) ↔ 𝑧 = 1)
54		0re 11141	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
55		fveqeq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((exp‘𝑥) = 𝑧 ↔ (exp‘0) = 𝑧))
56	55	rspcev 3562	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ (exp‘0) = 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
57	54, 56	mpan 697	. . . . . . . . . . 11 ⊢ ((exp‘0) = 𝑧 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
58	57	eqcoms 2749	. . . . . . . . . 10 ⊢ (𝑧 = (exp‘0) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
59	53, 58	sylbir 237	. . . . . . . . 9 ⊢ (𝑧 = 1 → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
60	59	adantl 483	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 𝑧 = 1) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
61		reeff1olem 26433	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
62	10, 61	sylan 587	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ⁺ ∧ 1 < 𝑧) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
63		1re 11139	. . . . . . . . 9 ⊢ 1 ∈ ℝ
64		lttri4 11225	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧))
65	10, 63, 64	sylancl 593	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧))
66	51, 60, 62, 65	mpjao3dan 1441	. . . . . . 7 ⊢ (𝑧 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
67		fvres 6850	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
68	67	eqeq1d 2743	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ (exp‘𝑥) = 𝑧))
69	68	rexbiia 3086	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑧)
70	66, 69	sylibr 236	. . . . . 6 ⊢ (𝑧 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ ((exp ↾ ℝ)‘𝑥) = 𝑧)
71		fvelrnb 6891	. . . . . . 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 236	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ran (exp ↾ ℝ))
74	73	ssriv 3921	. . . 4 ⊢ ℝ⁺ ⊆ ran (exp ↾ ℝ)
75	6, 74	eqssi 3933	. . 3 ⊢ ran (exp ↾ ℝ) = ℝ⁺
76		df-fo 6495	. . 3 ⊢ ((exp ↾ ℝ):ℝ–onto→ℝ⁺ ↔ ((exp ↾ ℝ) Fn ℝ ∧ ran (exp ↾ ℝ) = ℝ⁺))
77	4, 75, 76	mpbir2an 718	. 2 ⊢ (exp ↾ ℝ):ℝ–onto→ℝ⁺
78		df-f1o 6496	. 2 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ↔ ((exp ↾ ℝ):ℝ–1-1→ℝ⁺ ∧ (exp ↾ ℝ):ℝ–onto→ℝ⁺))
79	1, 77, 78	mpbir2an 718	1 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∨ w3o 1092 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 ⊆ wss 3885 class class class wbr 5075 ran crn 5622 ↾ cres 5623 Fn wfn 6484 ⟶wf 6485 –1-1→wf1 6486 –onto→wfo 6487 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 · cmul 11038 < clt 11174 -cneg 11373 / cdiv 11802 ℝ⁺crp 12937 expce 16021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-limc 25855 df-dv 25856

This theorem is referenced by: reefiso 26435 efcvx 26436 reefgim 26437 eff1olem 26534 dfrelog 26551 relogf1o 26552 dvrelog 26623

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