reeff1o - Metamath Proof Explorer

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Theorem reeff1o 26387

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 16033	. 2 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ⁺
2		f1f 6726	. . . 4 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
3		ffn 6658	. . . 4 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → (exp ↾ ℝ) Fn ℝ)
4	1, 2, 3	mp2b 10	. . 3 ⊢ (exp ↾ ℝ) Fn ℝ
5		frn 6665	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ⁺ → ran (exp ↾ ℝ) ⊆ ℝ⁺)
6	1, 2, 5	mp2b 10	. . . 4 ⊢ ran (exp ↾ ℝ) ⊆ ℝ⁺
7		elrp 12896	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧))
8		reclt1 12026	. . . . . . . . . . 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 12903	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
11		rpne0 12911	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ≠ 0)
12	10, 11	rereccld 11957	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → (1 / 𝑧) ∈ ℝ)
13		reeff1olem 26386	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 𝑧) = (exp‘𝑦) ↔ (exp‘𝑦) = (1 / 𝑧))
16		rpcnne0 12913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
17		recn 11105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
18		efcl 15993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (exp‘𝑦) ∈ ℂ)
20		efne0 16009	. . . . . . . . . . . . . . . . . . . . 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 11829	. . . . . . . . . . . . . . . . . . 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 16007	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → ((exp‘𝑦) · (exp‘-𝑦)) = 1)
26	25	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℂ → 1 = ((exp‘𝑦) · (exp‘-𝑦)))
27		negcl 11369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℂ → -𝑦 ∈ ℂ)
28		efcl 15993	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℂ → (exp‘-𝑦) ∈ ℂ)
30		ax-1cn 11073	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
31		divmul2 11789	. . . . . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . 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 11433	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
45		infm3lem 12089	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)
46		fveqeq2 6839	. . . . . . . . . . . . 13 ⊢ (𝑥 = -𝑦 → ((exp‘𝑥) = 𝑧 ↔ (exp‘-𝑦) = 𝑧))
47	44, 45, 46	rexxfr 5358	. . . . . . . . . . . 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 16002	. . . . . . . . . . 11 ⊢ (exp‘0) = 1
53	52	eqeq2i 2746	. . . . . . . . . 10 ⊢ (𝑧 = (exp‘0) ↔ 𝑧 = 1)
54		0re 11123	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
55		fveqeq2 6839	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((exp‘𝑥) = 𝑧 ↔ (exp‘0) = 𝑧))
56	55	rspcev 3573	. . . . . . . . . . . 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 2741	. . . . . . . . . 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 26386	. . . . . . . . 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 11121	. . . . . . . . 9 ⊢ 1 ∈ ℝ
64		lttri4 11206	. . . . . . . . 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 6849	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
68	67	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((exp ↾ ℝ)‘𝑥) = 𝑧 ↔ (exp‘𝑥) = 𝑧))
69	68	rexbiia 3078	. . . . . . 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 6890	. . . . . . 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 3934	. . . 4 ⊢ ℝ⁺ ⊆ ran (exp ↾ ℝ)
75	6, 74	eqssi 3947	. . 3 ⊢ ran (exp ↾ ℝ) = ℝ⁺
76		df-fo 6494	. . 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 6495	. 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 2113 ≠ wne 2929 ∃wrex 3057 ⊆ wss 3898 class class class wbr 5095 ran crn 5622 ↾ cres 5623 Fn wfn 6483 ⟶wf 6484 –1-1→wf1 6485 –onto→wfo 6486 –1-1-onto→wf1o 6487 ‘cfv 6488 (class class class)co 7354 ℂcc 11013 ℝcr 11014 0cc0 11015 1c1 11016 · cmul 11020 < clt 11155 -cneg 11354 / cdiv 11783 ℝ⁺crp 12894 expce 15972

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093 ax-addf 11094

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 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 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-pm 8761 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-fi 9304 df-sup 9335 df-inf 9336 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-q 12851 df-rp 12895 df-xneg 13015 df-xadd 13016 df-xmul 13017 df-ioo 13253 df-ico 13255 df-icc 13256 df-fz 13412 df-fzo 13559 df-fl 13700 df-seq 13913 df-exp 13973 df-fac 14185 df-bc 14214 df-hash 14242 df-shft 14978 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-limsup 15382 df-clim 15399 df-rlim 15400 df-sum 15598 df-ef 15978 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-starv 17180 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-unif 17188 df-hom 17189 df-cco 17190 df-rest 17330 df-topn 17331 df-0g 17349 df-gsum 17350 df-topgen 17351 df-pt 17352 df-prds 17355 df-xrs 17410 df-qtop 17415 df-imas 17416 df-xps 17418 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-submnd 18696 df-mulg 18985 df-cntz 19233 df-cmn 19698 df-psmet 21287 df-xmet 21288 df-met 21289 df-bl 21290 df-mopn 21291 df-fbas 21292 df-fg 21293 df-cnfld 21296 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22864 df-cld 22937 df-ntr 22938 df-cls 22939 df-nei 23016 df-lp 23054 df-perf 23055 df-cn 23145 df-cnp 23146 df-haus 23233 df-tx 23480 df-hmeo 23673 df-fil 23764 df-fm 23856 df-flim 23857 df-flf 23858 df-xms 24238 df-ms 24239 df-tms 24240 df-cncf 24801 df-limc 25797 df-dv 25798

This theorem is referenced by: reefiso 26388 efcvx 26389 reefgim 26390 eff1olem 26487 dfrelog 26504 relogf1o 26505 dvrelog 26576

W3C validator