Theorem reeff1olem 15872

Description: Lemma for reeff1o 15874. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Assertion

Ref	Expression
reeff1olem	|- ( ( U e. RR /\ 1 < U ) -> E. x e. RR ( exp ` x ) = U )

Distinct variable group:   x, U

Proof of Theorem reeff1olem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossicc 10361	. . 3 |- ( 0 (,) U ) C_ ( 0 [,] U )
2		0re 8326	. . . . 5 |- 0 e. RR
3		iccssre 10357	. . . . 5 |- ( ( 0 e. RR /\ U e. RR ) -> ( 0 [,] U ) C_ RR )
4	2, 3	mpan 428	. . . 4 |- ( U e. RR -> ( 0 [,] U ) C_ RR )
5	4	adantr 276	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ RR )
6	1, 5	sstrid 3259	. 2 |- ( ( U e. RR /\ 1 < U ) -> ( 0 (,) U ) C_ RR )
7	2	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> 0 e. RR )
8		simpl 109	. . 3 |- ( ( U e. RR /\ 1 < U ) -> U e. RR )
9		0lt1 8453	. . . . 5 |- 0 < 1
10		1re 8325	. . . . . 6 |- 1 e. RR
11		lttr 8399	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ U e. RR ) -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
12	2, 10, 11	mp3an12 1368	. . . . 5 |- ( U e. RR -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
13	9, 12	mpani 434	. . . 4 |- ( U e. RR -> ( 1 < U -> 0 < U ) )
14	13	imp 124	. . 3 |- ( ( U e. RR /\ 1 < U ) -> 0 < U )
15		ax-resscn 8271	. . . 4 |- RR C_ CC
16	5, 15	sstrdi 3260	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ CC )
17		efcn 15869	. . . 4 |- exp e. ( CC -cn-> CC )
18	17	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> exp e. ( CC -cn-> CC ) )
19		ssel2 3243	. . . . 5 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> y e. RR )
20	19	reefcld 12436	. . . 4 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> ( exp ` y ) e. RR )
21	5, 20	sylan 283	. . 3 |- ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) -> ( exp ` y ) e. RR )
22		ef0 12439	. . . . 5 |- ( exp ` 0 ) = 1
23		simpr 110	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> 1 < U )
24	22, 23	eqbrtrid 4165	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` 0 ) < U )
25		peano2re 8462	. . . . . 6 |- ( U e. RR -> ( U + 1 ) e. RR )
26	25	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) e. RR )
27		reefcl 12435	. . . . . 6 |- ( U e. RR -> ( exp ` U ) e. RR )
28	27	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` U ) e. RR )
29		ltp1 9174	. . . . . 6 |- ( U e. RR -> U < ( U + 1 ) )
30	29	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> U < ( U + 1 ) )
31	8	recnd 8354	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. CC )
32		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
33		addcom 8463	. . . . . . 7 |- ( ( U e. CC /\ 1 e. CC ) -> ( U + 1 ) = ( 1 + U ) )
34	31, 32, 33	sylancl 417	. . . . . 6 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) = ( 1 + U ) )
35	8, 14	elrpd 10094	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. RR+ )
36		efgt1p 12463	. . . . . . 7 |- ( U e. RR+ -> ( 1 + U ) < ( exp ` U ) )
37	35, 36	syl 14	. . . . . 6 |- ( ( U e. RR /\ 1 < U ) -> ( 1 + U ) < ( exp ` U ) )
38	34, 37	eqbrtrd 4152	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) < ( exp ` U ) )
39	8, 26, 28, 30, 38	lttrd 8452	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> U < ( exp ` U ) )
40	24, 39	jca 306	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( ( exp ` 0 ) < U /\ U < ( exp ` U ) ) )
41		simplll 539	. . . . . . 7 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> U e. RR )
42	2, 41, 3	sylancr 418	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( 0 [,] U ) C_ RR )
43		simplr 533	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. ( 0 [,] U ) )
44	42, 43	sseldd 3249	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. RR )
45		simprl 535	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. ( 0 [,] U ) )
46	42, 45	sseldd 3249	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. RR )
47	44, 46	jca 306	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( y e. RR /\ z e. RR ) )
48		simprr 537	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y < z )
49		efltim 12465	. . . 4 |- ( ( y e. RR /\ z e. RR ) -> ( y < z -> ( exp ` y ) < ( exp ` z ) ) )
50	47, 48, 49	sylc 62	. . 3 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( exp ` y ) < ( exp ` z ) )
51	7, 8, 8, 14, 16, 18, 21, 40, 50	ivthinc 15744	. 2 |- ( ( U e. RR /\ 1 < U ) -> E. x e. ( 0 (,) U ) ( exp ` x ) = U )
52		ssrexv 3313	. 2 |- ( ( 0 (,) U ) C_ RR -> ( E. x e. ( 0 (,) U ) ( exp ` x ) = U -> E. x e. RR ( exp ` x ) = U ) )
53	6, 51, 52	sylc 62	1 |- ( ( U e. RR /\ 1 < U ) -> E. x e. RR ( exp ` x ) = U )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   RR+crp 10054   (,)cioo 10290   [,]cicc 10293   expce 12409   -cn->ccncf 15671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758

This theorem is used by:  reeff1oleme  15873  reeff1o  15874