Theorem reeff1 7350

Description: The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.)

Assertion

Ref	Expression
reeff1	|- (exp |` RR):RR-1-1->(0(,) +oo)

Proof of Theorem reeff1

Step	Hyp	Ref	Expression
1		f1fv 3859	. 2 |- ((exp |` RR):RR-1-1->(0(,) +oo) <-> ((exp |` RR):RR-->(0(,) +oo) /\ A.x e. RR A.y e. RR (((exp |` RR)` x) = ((exp |` RR)` y) -> x = y)))
2		df-ef 7240	. . . . 5 |- exp = {<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))}
3		reseq1 3352	. . . . 5 |- (exp = {<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))} -> (exp |` RR) = ({<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))} |` RR))
4	2, 3	ax-mp 7	. . . 4 |- (exp |` RR) = ({<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))} |` RR)
5		axresscn 5240	. . . . 5 |- RR (_ CC
6		resopab2 3382	. . . . 5 |- (RR (_ CC -> ({<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))} |` RR) = {<.x, y>. | (x e. RR /\ y = sum_k e. NN0 ((x^k) / (!` k)))})
7	5, 6	ax-mp 7	. . . 4 |- ({<.x, y>. | (x e. CC /\ y = sum_k e. NN0 ((x^k) / (!` k)))} |` RR) = {<.x, y>. | (x e. RR /\ y = sum_k e. NN0 ((x^k) / (!` k)))}
8	4, 7	eqtr 1487	. . 3 |- (exp |` RR) = {<.x, y>. | (x e. RR /\ y = sum_k e. NN0 ((x^k) / (!` k)))}
9		recnt 5285	. . . . . . 7 |- (x e. RR -> x e. CC)
10		efvalt 7250	. . . . . . 7 |- (x e. CC -> (exp` x) = sum_k e. NN0 ((x^k) / (!` k)))
11	9, 10	syl 10	. . . . . 6 |- (x e. RR -> (exp` x) = sum_k e. NN0 ((x^k) / (!` k)))
12		reefclt 7260	. . . . . 6 |- (x e. RR -> (exp` x) e. RR)
13	11, 12	eqeltrrd 1541	. . . . 5 |- (x e. RR -> sum_k e. NN0 ((x^k) / (!` k)) e. RR)
14		efgt0t 7346	. . . . . 6 |- (x e. RR -> 0 < (exp` x))
15	14, 11	breqtrd 2629	. . . . 5 |- (x e. RR -> 0 < sum_k e. NN0 ((x^k) / (!` k)))
16	13, 15	jca 288	. . . 4 |- (x e. RR -> (sum_k e. NN0 ((x^k) / (!` k)) e. RR /\ 0 < sum_k e. NN0 ((x^k) / (!` k))))
17		repos 6332	. . . 4 |- (sum_k e. NN0 ((x^k) / (!` k)) e. (0(,) +oo) <-> (sum_k e. NN0 ((x^k) / (!` k)) e. RR /\ 0 < sum_k e. NN0 ((x^k) / (!` k))))
18	16, 17	sylibr 200	. . 3 |- (x e. RR -> sum_k e. NN0 ((x^k) / (!` k)) e. (0(,) +oo))
19	8, 18	fopab 3812	. 2 |- (exp |` RR):RR-->(0(,) +oo)
20		fvres 3719	. . . . 5 |- (x e. RR -> ((exp |` RR)` x) = (exp` x))
21		fvres 3719	. . . . 5 |- (y e. RR -> ((exp |` RR)` y) = (exp` y))
22	20, 21	eqeqan12d 1482	. . . 4 |- ((x e. RR /\ y e. RR) -> (((exp |` RR)` x) = ((exp |` RR)` y) <-> (exp` x) = (exp` y)))
23		fveq2 3709	. . . . . . . 8 |- (x = if(x e. RR, x, 1) -> (exp` x) = (exp` if(x e. RR, x, 1)))
24	23	eqeq1d 1475	. . . . . . 7 |- (x = if(x e. RR, x, 1) -> ((exp` x) = (exp` y) <-> (exp` if(x e. RR, x, 1)) = (exp` y)))
25		eqeq1 1473	. . . . . . 7 |- (x = if(x e. RR, x, 1) -> (x = y <-> if(x e. RR, x, 1) = y))
26	24, 25	bibi12d 627	. . . . . 6 |- (x = if(x e. RR, x, 1) -> (((exp` x) = (exp` y) <-> x = y) <-> ((exp` if(x e. RR, x, 1)) = (exp` y) <-> if(x e. RR, x, 1) = y)))
27		fveq2 3709	. . . . . . . 8 |- (y = if(y e. RR, y, 1) -> (exp` y) = (exp` if(y e. RR, y, 1)))
28	27	eqeq2d 1478	. . . . . . 7 |- (y = if(y e. RR, y, 1) -> ((exp` if(x e. RR, x, 1)) = (exp` y) <-> (exp` if(x e. RR, x, 1)) = (exp` if(y e. RR, y, 1))))
29		eqeq2 1476	. . . . . . 7 |- (y = if(y e. RR, y, 1) -> (if(x e. RR, x, 1) = y <-> if(x e. RR, x, 1) = if(y e. RR, y, 1)))
30	28, 29	bibi12d 627	. . . . . 6 |- (y = if(y e. RR, y, 1) -> (((exp` if(x e. RR, x, 1)) = (exp` y) <-> if(x e. RR, x, 1) = y) <-> ((exp` if(x e. RR, x, 1)) = (exp` if(y e. RR, y, 1)) <-> if(x e. RR, x, 1) = if(y e. RR, y, 1))))
31		1re 5407	. . . . . . . 8 |- 1 e. RR
32	31	elimel 2384	. . . . . . 7 |- if(x e. RR, x, 1) e. RR
33	31	elimel 2384	. . . . . . 7 |- if(y e. RR, y, 1) e. RR
34	32, 33	reef11 7349	. . . . . 6 |- ((exp` if(x e. RR, x, 1)) = (exp` if(y e. RR, y, 1)) <-> if(x e. RR, x, 1) = if(y e. RR, y, 1))
35	26, 30, 34	dedth2h 2377	. . . . 5 |- ((x e. RR /\ y e. RR) -> ((exp` x) = (exp` y) <-> x = y))
36	35	biimpd 153	. . . 4 |- ((x e. RR /\ y e. RR) -> ((exp` x) = (exp` y) -> x = y))
37	22, 36	sylbid 203	. . 3 |- ((x e. RR /\ y e. RR) -> (((exp |` RR)` x) = ((exp |` RR)` y) -> x = y))
38	37	rgen2a 1691	. 2 |- A.x e. RR A.y e. RR (((exp |` RR)` x) = ((exp |` RR)` y) -> x = y)
39	1, 19, 38	mpbir2an 728	1 |- (exp |` RR):RR-1-1->(0(,) +oo)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   (_ wss 2037  ifcif 2351   class class class wbr 2609  {copab 2656   |` cres 3162  -->wf 3168  -1-1->wf1 3169  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   / cdiv 5266  NN0cn0 5269   +oocpnf 5455   < clt 5458  (,)cioo 6294  ^cexp 6500  !cfa 6868  sum_csu 6917  expce 7235

This theorem is referenced by:  reeff1o 7368  eff1i 8665

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-fl 6172  df-seq1 6245  df-shft 6278  df-ioo 6298  df-uz 6350  df-fz 6400  df-seqz 6465  df-seq0 6466  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-fac 6869  df-bc 6894  df-clim 6913  df-sum 6918  df-ef 7240

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