reefiso - Metamath Proof Explorer

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Theorem reefiso 7519

Description: The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)

**Assertion**
Ref	Expression
reefiso	⊢ (exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺)

**Proof of Theorem reefiso**
Step	Hyp	Ref	Expression
1		df-iso 3256	. 2 ⊢ ((exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺) ↔ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ⋀ ∀x ∈ ℝ ∀y ∈ ℝ (x < y ↔ ((exp ↾ ℝ) ‘x) < ((exp ↾ ℝ) ‘y))))
2		reeff1o2 7518	. 2 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
3		breq1 2677	. . . . 5 ⊢ (x = if(x ∈ ℝ, x, 0) → (x < y ↔ if(x ∈ ℝ, x, 0) < y))
4		fveq2 3781	. . . . . 6 ⊢ (x = if(x ∈ ℝ, x, 0) → ((exp ↾ ℝ) ‘x) = ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)))
5	4	breq1d 2684	. . . . 5 ⊢ (x = if(x ∈ ℝ, x, 0) → (((exp ↾ ℝ) ‘x) < ((exp ↾ ℝ) ‘y) ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘y)))
6	3, 5	bibi12d 640	. . . 4 ⊢ (x = if(x ∈ ℝ, x, 0) → ((x < y ↔ ((exp ↾ ℝ) ‘x) < ((exp ↾ ℝ) ‘y)) ↔ ( if(x ∈ ℝ, x, 0) < y ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘y))))
7		breq2 2678	. . . . 5 ⊢ (y = if(y ∈ ℝ, y, 0) → ( if(x ∈ ℝ, x, 0) < y ↔ if(x ∈ ℝ, x, 0) < if(y ∈ ℝ, y, 0)))
8		fveq2 3781	. . . . . 6 ⊢ (y = if(y ∈ ℝ, y, 0) → ((exp ↾ ℝ) ‘y) = ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)))
9	8	breq2d 2685	. . . . 5 ⊢ (y = if(y ∈ ℝ, y, 0) → (((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘y) ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0))))
10	7, 9	bibi12d 640	. . . 4 ⊢ (y = if(y ∈ ℝ, y, 0) → (( if(x ∈ ℝ, x, 0) < y ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘y)) ↔ ( if(x ∈ ℝ, x, 0) < if(y ∈ ℝ, y, 0) ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)))))
11		0re 5505	. . . . . . 7 ⊢ 0 ∈ ℝ
12	11	elimel 2446	. . . . . 6 ⊢ if(x ∈ ℝ, x, 0) ∈ ℝ
13	11	elimel 2446	. . . . . 6 ⊢ if(y ∈ ℝ, y, 0) ∈ ℝ
14	12, 13	efltbi 7498	. . . . 5 ⊢ ( if(x ∈ ℝ, x, 0) < if(y ∈ ℝ, y, 0) ↔ (exp ‘ if(x ∈ ℝ, x, 0)) < (exp ‘ if(y ∈ ℝ, y, 0)))
15		fvres 3791	. . . . . . 7 ⊢ ( if(x ∈ ℝ, x, 0) ∈ ℝ → ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) = (exp ‘ if(x ∈ ℝ, x, 0)))
16	12, 15	ax-mp 7	. . . . . 6 ⊢ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) = (exp ‘ if(x ∈ ℝ, x, 0))
17		fvres 3791	. . . . . . 7 ⊢ ( if(y ∈ ℝ, y, 0) ∈ ℝ → ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)) = (exp ‘ if(y ∈ ℝ, y, 0)))
18	13, 17	ax-mp 7	. . . . . 6 ⊢ ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)) = (exp ‘ if(y ∈ ℝ, y, 0))
19	16, 18	breq12i 2683	. . . . 5 ⊢ (((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)) ↔ (exp ‘ if(x ∈ ℝ, x, 0)) < (exp ‘ if(y ∈ ℝ, y, 0)))
20	14, 19	bitr4i 183	. . . 4 ⊢ ( if(x ∈ ℝ, x, 0) < if(y ∈ ℝ, y, 0) ↔ ((exp ↾ ℝ) ‘ if(x ∈ ℝ, x, 0)) < ((exp ↾ ℝ) ‘ if(y ∈ ℝ, y, 0)))
21	6, 10, 20	dedth2h 2439	. . 3 ⊢ ((x ∈ ℝ ⋀ y ∈ ℝ) → (x < y ↔ ((exp ↾ ℝ) ‘x) < ((exp ↾ ℝ) ‘y)))
22	21	rgen2a 1746	. 2 ⊢ ∀x ∈ ℝ ∀y ∈ ℝ (x < y ↔ ((exp ↾ ℝ) ‘x) < ((exp ↾ ℝ) ‘y))
23	1, 2, 22	mpbir2an 742	1 ⊢ (exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺)

Colors of variables: wff set class

Syntax hints: ↔ wb 153 = wceq 997 ∈ wcel 999 ∀wral 1692 ifcif 2413 class class class wbr 2674 ↾ cres 3229 –1-1-onto→wf1o 3238 ‘cfv 3239 Isom wiso 3240 ℝcr 5298 0cc0 5299 ℝ⁺crp 5365 < clt 5551 expce 7383

This theorem is referenced by: relogiso 8858

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1003 ax-gen 1004 ax-8 1005 ax-9 1006 ax-10 1007 ax-11 1008 ax-12 1009 ax-13 1010 ax-14 1011 ax-17 1012 ax-4 1014 ax-5o 1016 ax-6o 1019 ax-9o 1164 ax-10o 1182 ax-16 1252 ax-11o 1260 ax-ext 1504 ax-rep 2748 ax-sep 2758 ax-nul 2765 ax-pow 2798 ax-pr 2835 ax-un 2922 ax-inf2 4687

This theorem depends on definitions: df-bi 154 df-or 231 df-an 232 df-3or 788 df-3an 789 df-ex 1022 df-sb 1214 df-eu 1424 df-mo 1425 df-clab 1510 df-cleq 1515 df-clel 1518 df-ne 1634 df-nel 1635 df-ral 1696 df-rex 1697 df-reu 1698 df-rab 1699 df-v 1859 df-sbc 1989 df-csb 2052 df-dif 2100 df-un 2101 df-in 2102 df-ss 2104 df-pss 2106 df-nul 2332 df-if 2414 df-pw 2454 df-sn 2464 df-pr 2465 df-tp 2467 df-op 2468 df-uni 2558 df-int 2588 df-iun 2622 df-br 2675 df-opab 2722 df-tr 2736 df-eprel 2888 df-id 2891 df-po 2896 df-so 2906 df-fr 2974 df-we 2991 df-ord 3008 df-on 3009 df-lim 3010 df-suc 3011 df-om 3189 df-xp 3241 df-rel 3242 df-cnv 3243 df-co 3244 df-dm 3245 df-rn 3246 df-res 3247 df-ima 3248 df-fun 3249 df-fn 3250 df-f 3251 df-f1 3252 df-fo 3253 df-f1o 3254 df-fv 3255 df-iso 3256 df-rdg 3990 df-opr 4023 df-oprab 4024 df-1st 4137 df-2nd 4138 df-1o 4191 df-oadd 4193 df-omul 4194 df-er 4319 df-ec 4321 df-qs 4324 df-en 4429 df-dom 4430 df-sdom 4431 df-sup 4634 df-ni 5065 df-pli 5066 df-mi 5067 df-lti 5068 df-plpq 5100 df-mpq 5101 df-enq 5102 df-nq 5103 df-plq 5104 df-mq 5105 df-rq 5106 df-ltq 5107 df-1q 5108 df-np 5151 df-1p 5152 df-plp 5153 df-mp 5154 df-ltp 5155 df-plpr 5229 df-mpr 5230 df-enr 5231 df-nr 5232 df-plr 5233 df-mr 5234 df-ltr 5235 df-0r 5236 df-1r 5237 df-m1r 5238 df-c 5305 df-0 5306 df-1 5307 df-i 5308 df-r 5309 df-plus 5310 df-mul 5311 df-lt 5312 df-sub 5421 df-neg 5423 df-pnf 5552 df-mnf 5553 df-xr 5554 df-ltxr 5555 df-le 5556 df-div 5768 df-n 5985 df-2 6031 df-3 6032 df-4 6033 df-rp 6106 df-n0 6182 df-z 6218 df-q 6308 df-fl 6335 df-ioo 6386 df-icc 6389 df-uz 6444 df-fz 6494 df-seq1 6567 df-shft 6600 df-seqz 6622 df-seq0 6623 df-exp 6658 df-sqr 6760 df-re 6841 df-im 6842 df-cj 6843 df-abs 6844 df-fac 7022 df-bc 7047 df-clim 7065 df-sum 7070 df-cncf 7353 df-ef 7388