Theorem reeff1olem 13031

Description: Lemma for reeff1o 13033. (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 9841	. . 3 |- ( 0 (,) U ) C_ ( 0 [,] U )
2		0re 7857	. . . . 5 |- 0 e. RR
3		iccssre 9837	. . . . 5 |- ( ( 0 e. RR /\ U e. RR ) -> ( 0 [,] U ) C_ RR )
4	2, 3	mpan 421	. . . 4 |- ( U e. RR -> ( 0 [,] U ) C_ RR )
5	4	adantr 274	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ RR )
6	1, 5	sstrid 3135	. 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 108	. . 3 |- ( ( U e. RR /\ 1 < U ) -> U e. RR )
9		0lt1 7981	. . . . 5 |- 0 < 1
10		1re 7856	. . . . . 6 |- 1 e. RR
11		lttr 7930	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ U e. RR ) -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
12	2, 10, 11	mp3an12 1306	. . . . 5 |- ( U e. RR -> ( ( 0 < 1 /\ 1 < U ) -> 0 < U ) )
13	9, 12	mpani 427	. . . 4 |- ( U e. RR -> ( 1 < U -> 0 < U ) )
14	13	imp 123	. . 3 |- ( ( U e. RR /\ 1 < U ) -> 0 < U )
15		ax-resscn 7803	. . . 4 |- RR C_ CC
16	5, 15	sstrdi 3136	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( 0 [,] U ) C_ CC )
17		efcn 13028	. . . 4 |- exp e. ( CC -cn-> CC )
18	17	a1i 9	. . 3 |- ( ( U e. RR /\ 1 < U ) -> exp e. ( CC -cn-> CC ) )
19		ssel2 3119	. . . . 5 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> y e. RR )
20	19	reefcld 11543	. . . 4 |- ( ( ( 0 [,] U ) C_ RR /\ y e. ( 0 [,] U ) ) -> ( exp ` y ) e. RR )
21	5, 20	sylan 281	. . 3 |- ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) -> ( exp ` y ) e. RR )
22		ef0 11546	. . . . 5 |- ( exp ` 0 ) = 1
23		simpr 109	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> 1 < U )
24	22, 23	eqbrtrid 3995	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` 0 ) < U )
25		peano2re 7990	. . . . . 6 |- ( U e. RR -> ( U + 1 ) e. RR )
26	25	adantr 274	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) e. RR )
27		reefcl 11542	. . . . . 6 |- ( U e. RR -> ( exp ` U ) e. RR )
28	27	adantr 274	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( exp ` U ) e. RR )
29		ltp1 8694	. . . . . 6 |- ( U e. RR -> U < ( U + 1 ) )
30	29	adantr 274	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> U < ( U + 1 ) )
31	8	recnd 7885	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. CC )
32		ax-1cn 7804	. . . . . . 7 |- 1 e. CC
33		addcom 7991	. . . . . . 7 |- ( ( U e. CC /\ 1 e. CC ) -> ( U + 1 ) = ( 1 + U ) )
34	31, 32, 33	sylancl 410	. . . . . 6 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) = ( 1 + U ) )
35	8, 14	elrpd 9578	. . . . . . 7 |- ( ( U e. RR /\ 1 < U ) -> U e. RR+ )
36		efgt1p 11570	. . . . . . 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 3982	. . . . 5 |- ( ( U e. RR /\ 1 < U ) -> ( U + 1 ) < ( exp ` U ) )
39	8, 26, 28, 30, 38	lttrd 7980	. . . 4 |- ( ( U e. RR /\ 1 < U ) -> U < ( exp ` U ) )
40	24, 39	jca 304	. . 3 |- ( ( U e. RR /\ 1 < U ) -> ( ( exp ` 0 ) < U /\ U < ( exp ` U ) ) )
41		simplll 523	. . . . . . 7 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> U e. RR )
42	2, 41, 3	sylancr 411	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( 0 [,] U ) C_ RR )
43		simplr 520	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. ( 0 [,] U ) )
44	42, 43	sseldd 3125	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y e. RR )
45		simprl 521	. . . . . 6 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. ( 0 [,] U ) )
46	42, 45	sseldd 3125	. . . . 5 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> z e. RR )
47	44, 46	jca 304	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> ( y e. RR /\ z e. RR ) )
48		simprr 522	. . . 4 |- ( ( ( ( U e. RR /\ 1 < U ) /\ y e. ( 0 [,] U ) ) /\ ( z e. ( 0 [,] U ) /\ y < z ) ) -> y < z )
49		efltim 11572	. . . 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 12960	. 2 |- ( ( U e. RR /\ 1 < U ) -> E. x e. ( 0 (,) U ) ( exp ` x ) = U )
52		ssrexv 3189	. 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 103   = wceq 1332   e. wcel 2125   E.wrex 2433   C_ wss 3098   class class class wbr 3961   ` cfv 5163  (class class class)co 5814   CCcc 7709   RRcr 7710   0cc0 7711   1c1 7712   + caddc 7714   < clt 7891   RR+crp 9538   (,)cioo 9770   [,]cicc 9773   expce 11516   -cn->ccncf 12896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831  ax-pre-suploc 7832  ax-addf 7833  ax-mulf 7834

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-disj 3939  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-of 6022  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-frec 6328  df-1o 6353  df-oadd 6357  df-er 6469  df-map 6584  df-pm 6585  df-en 6675  df-dom 6676  df-fin 6677  df-sup 6916  df-inf 6917  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-xneg 9657  df-xadd 9658  df-ioo 9774  df-ico 9776  df-icc 9777  df-fz 9891  df-fzo 10020  df-seqfrec 10323  df-exp 10397  df-fac 10577  df-bc 10599  df-ihash 10627  df-shft 10692  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-clim 11153  df-sumdc 11228  df-ef 11522  df-rest 12292  df-topgen 12311  df-psmet 12326  df-xmet 12327  df-met 12328  df-bl 12329  df-mopn 12330  df-top 12335  df-topon 12348  df-bases 12380  df-ntr 12435  df-cn 12527  df-cnp 12528  df-tx 12592  df-cncf 12897  df-limced 12964  df-dvap 12965

This theorem is referenced by:  reeff1oleme  13032  reeff1o  13033