Theorem reeff1olem 15358

Description: Lemma for reeff1o 15360. (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 10116	. . 3 |- ( 0 (,) U ) C_ ( 0 [,] U )
2		0re 8107	. . . . 5 |- 0 e. RR
3		iccssre 10112	. . . . 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 3212	. 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 8234	. . . . 5 |- 0 < 1
10		1re 8106	. . . . . 6 |- 1 e. RR
11		lttr 8181	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ U e. RR ) -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
12	2, 10, 11	mp3an12 1340	. . . . 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 8052	. . . 4 |- RR C_ CC
16	5, 15	sstrdi 3213	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ CC )
17		efcn 15355	. . . 4 |- exp e. ( CC -cn-> CC )
18	17	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> exp e. ( CC -cn-> CC ) )
19		ssel2 3196	. . . . 5 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> y e. RR )
20	19	reefcld 12095	. . . 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 12098	. . . . 5 |- ( exp ` 0 ) = 1
23		simpr 110	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> 1 < U )
24	22, 23	eqbrtrid 4094	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` 0 ) < U )
25		peano2re 8243	. . . . . 6 |- ( U e. RR -> ( U + 1 ) e. RR )
26	25	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) e. RR )
27		reefcl 12094	. . . . . 6 |- ( U e. RR -> ( exp ` U ) e. RR )
28	27	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` U ) e. RR )
29		ltp1 8952	. . . . . 6 |- ( U e. RR -> U < ( U + 1 ) )
30	29	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> U < ( U + 1 ) )
31	8	recnd 8136	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. CC )
32		ax-1cn 8053	. . . . . . 7 |- 1 e. CC
33		addcom 8244	. . . . . . 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 9850	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. RR+ )
36		efgt1p 12122	. . . . . . 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 4081	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) < ( exp ` U ) )
39	8, 26, 28, 30, 38	lttrd 8233	. . . 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 533	. . . . . . 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 528	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. ( 0 [,] U ) )
44	42, 43	sseldd 3202	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. RR )
45		simprl 529	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. ( 0 [,] U ) )
46	42, 45	sseldd 3202	. . . . 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 531	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y < z )
49		efltim 12124	. . . 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 15230	. 2 |- ( ( U e. RR /\ 1 < U ) -> E. x e. ( 0 (,) U ) ( exp ` x ) = U )
52		ssrexv 3266	. 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 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   < clt 8142   RR+crp 9810   (,)cioo 10045   [,]cicc 10048   expce 12068   -cn->ccncf 15157

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080  ax-pre-suploc 8081  ax-addf 8082  ax-mulf 8083

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-disj 4036  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-map 6760  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-ico 10051  df-icc 10052  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-bc 10930  df-ihash 10958  df-shft 11241  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780  df-ef 12074  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-tx 14840  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by:  reeff1oleme  15359  reeff1o  15360