Theorem reeff1olem 14947

Description: Lemma for reeff1o 14949. (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 10028	. . 3 |- ( 0 (,) U ) C_ ( 0 [,] U )
2		0re 8021	. . . . 5 |- 0 e. RR
3		iccssre 10024	. . . . 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 3191	. 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 8148	. . . . 5 |- 0 < 1
10		1re 8020	. . . . . 6 |- 1 e. RR
11		lttr 8095	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ U e. RR ) -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
12	2, 10, 11	mp3an12 1338	. . . . 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 7966	. . . 4 |- RR C_ CC
16	5, 15	sstrdi 3192	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ CC )
17		efcn 14944	. . . 4 |- exp e. ( CC -cn-> CC )
18	17	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> exp e. ( CC -cn-> CC ) )
19		ssel2 3175	. . . . 5 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> y e. RR )
20	19	reefcld 11815	. . . 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 11818	. . . . 5 |- ( exp ` 0 ) = 1
23		simpr 110	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> 1 < U )
24	22, 23	eqbrtrid 4065	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` 0 ) < U )
25		peano2re 8157	. . . . . 6 |- ( U e. RR -> ( U + 1 ) e. RR )
26	25	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) e. RR )
27		reefcl 11814	. . . . . 6 |- ( U e. RR -> ( exp ` U ) e. RR )
28	27	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` U ) e. RR )
29		ltp1 8865	. . . . . 6 |- ( U e. RR -> U < ( U + 1 ) )
30	29	adantr 276	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> U < ( U + 1 ) )
31	8	recnd 8050	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. CC )
32		ax-1cn 7967	. . . . . . 7 |- 1 e. CC
33		addcom 8158	. . . . . . 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 9762	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. RR+ )
36		efgt1p 11842	. . . . . . 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 4052	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) < ( exp ` U ) )
39	8, 26, 28, 30, 38	lttrd 8147	. . . 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 3181	. . . . 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 3181	. . . . 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 11844	. . . 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 14822	. 2 |- ( ( U e. RR /\ 1 < U ) -> E. x e. ( 0 (,) U ) ( exp ` x ) = U )
52		ssrexv 3245	. 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 1364   e. wcel 2164   E.wrex 2473   C_ wss 3154   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   < clt 8056   RR+crp 9722   (,)cioo 9957   [,]cicc 9960   expce 11788   -cn->ccncf 14749

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994  ax-pre-suploc 7995  ax-addf 7996  ax-mulf 7997

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-disj 4008  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-map 6706  df-pm 6707  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-ioo 9961  df-ico 9963  df-icc 9964  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-bc 10822  df-ihash 10850  df-shft 10962  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794  df-rest 12855  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-ntr 14275  df-cn 14367  df-cnp 14368  df-tx 14432  df-cncf 14750  df-limced 14835  df-dvap 14836

This theorem is referenced by:  reeff1oleme  14948  reeff1o  14949