Theorem reeff1olem 15768

Description: Lemma for reeff1o 15770. (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 10316	. . 3 |- ( 0 (,) U ) C_ ( 0 [,] U )
2		0re 8292	. . . . 5 |- 0 e. RR
3		iccssre 10312	. . . . 5 |- ( ( 0 e. RR /\ U e. RR ) -> ( 0 [,] U ) C_ RR )
4	2, 3	mpan 424	. . . 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 3253	. 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 8419	. . . . 5 |- 0 < 1
10		1re 8291	. . . . . 6 |- 1 e. RR
11		lttr 8365	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ U e. RR ) -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
12	2, 10, 11	mp3an12 1364	. . . . 5 |- ( U e. RR -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
13	9, 12	mpani 430	. . . 4 |- ( U e. RR -> ( 1 < U -> 0 < U ) )
14	13	imp 124	. . 3 |- ( ( U e. RR /\ 1 < U ) -> 0 < U )
15		ax-resscn 8237	. . . 4 |- RR C_ CC
16	5, 15	sstrdi 3254	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ CC )
17		efcn 15765	. . . 4 |- exp e. ( CC -cn-> CC )
18	17	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> exp e. ( CC -cn-> CC ) )
19		ssel2 3237	. . . . 5 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> y e. RR )
20	19	reefcld 12386	. . . 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 12389	. . . . 5 |- ( exp ` 0 ) = 1
23		simpr 110	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> 1 < U )
24	22, 23	eqbrtrid 4150	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` 0 ) < U )
25		peano2re 8428	. . . . . 6 |- ( U e. RR -> ( U + 1 ) e. RR )
26	25	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) e. RR )
27		reefcl 12385	. . . . . 6 |- ( U e. RR -> ( exp ` U ) e. RR )
28	27	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` U ) e. RR )
29		ltp1 9140	. . . . . 6 |- ( U e. RR -> U < ( U + 1 ) )
30	29	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> U < ( U + 1 ) )
31	8	recnd 8320	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. CC )
32		ax-1cn 8238	. . . . . . 7 |- 1 e. CC
33		addcom 8429	. . . . . . 7 |- ( ( U e. CC /\ 1 e. CC ) -> ( U + 1 ) = ( 1 + U ) )
34	31, 32, 33	sylancl 413	. . . . . 6 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) = ( 1 + U ) )
35	8, 14	elrpd 10049	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. RR+ )
36		efgt1p 12413	. . . . . . 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 4137	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) < ( exp ` U ) )
39	8, 26, 28, 30, 38	lttrd 8418	. . . 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 535	. . . . . . 7 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> U e. RR )
42	2, 41, 3	sylancr 414	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( 0 [,] U ) C_ RR )
43		simplr 529	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. ( 0 [,] U ) )
44	42, 43	sseldd 3243	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. RR )
45		simprl 531	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. ( 0 [,] U ) )
46	42, 45	sseldd 3243	. . . . 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 533	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y < z )
49		efltim 12415	. . . 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 15640	. 2 |- ( ( U e. RR /\ 1 < U ) -> E. x e. ( 0 (,) U ) ( exp ` x ) = U )
52		ssrexv 3307	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3214   class class class wbr 4115   ` cfv 5359  (class class class)co 6060   CCcc 8143   RRcr 8144   0cc0 8145   1c1 8146   + caddc 8148   < clt 8326   RR+crp 10009   (,)cioo 10245   [,]cicc 10248   expce 12359   -cn->ccncf 15567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263  ax-arch 8264  ax-caucvg 8265  ax-pre-suploc 8266  ax-addf 8267  ax-mulf 8268

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-disj 4092  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-isom 5368  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-of 6277  df-1st 6349  df-2nd 6350  df-recs 6551  df-irdg 6616  df-frec 6637  df-1o 6662  df-oadd 6666  df-er 6782  df-map 6899  df-pm 6900  df-en 6991  df-dom 6992  df-fin 6993  df-sup 7290  df-inf 7291  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-n0 9519  df-z 9600  df-uz 9877  df-q 9975  df-rp 10010  df-xneg 10129  df-xadd 10130  df-ioo 10249  df-ico 10251  df-icc 10252  df-fz 10367  df-fzo 10504  df-seqfrec 10839  df-exp 10930  df-fac 11118  df-bc 11140  df-ihash 11169  df-shft 11530  df-cj 11557  df-re 11558  df-im 11559  df-rsqrt 11714  df-abs 11715  df-clim 11995  df-sumdc 12070  df-ef 12365  df-rest 13544  df-topgen 13563  df-psmet 14823  df-xmet 14824  df-met 14825  df-bl 14826  df-mopn 14827  df-top 14995  df-topon 15008  df-bases 15040  df-ntr 15093  df-cn 15185  df-cnp 15186  df-tx 15250  df-cncf 15568  df-limced 15653  df-dvap 15654

This theorem is referenced by:  reeff1oleme  15769  reeff1o  15770