Theorem reeff1oleme 15440

Description: Lemma for reeff1o 15441. (Contributed by Jim Kingdon, 15-May-2024.)

Assertion

Ref	Expression
reeff1oleme	|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )

Distinct variable group:   x, U

Proof of Theorem reeff1oleme

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ere 12176	. . . . 5 |- _e e. RR
2	1	a1i 9	. . . 4 |- ( U e. ( 0 (,) _e ) -> _e e. RR )
3		elioore 10104	. . . 4 |- ( U e. ( 0 (,) _e ) -> U e. RR )
4		0xr 8189	. . . . . . 7 |- 0 e. RR*
5	1	rexri 8200	. . . . . . 7 |- _e e. RR*
6		elioo2 10113	. . . . . . 7 |- ( ( 0 e. RR* /\ _e e. RR* ) -> ( U e. ( 0 (,) _e ) <-> ( U e. RR /\ 0 < U /\ U < _e ) ) )
7	4, 5, 6	mp2an 426	. . . . . 6 |- ( U e. ( 0 (,) _e ) <-> ( U e. RR /\ 0 < U /\ U < _e ) )
8	7	simp2bi 1037	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 0 < U )
9	3, 8	gt0ap0d 8772	. . . 4 |- ( U e. ( 0 (,) _e ) -> U # 0 )
10	2, 3, 9	redivclapd 8978	. . 3 |- ( U e. ( 0 (,) _e ) -> ( _e / U ) e. RR )
11	3	recnd 8171	. . . . . 6 |- ( U e. ( 0 (,) _e ) -> U e. CC )
12	11	mulid2d 8161	. . . . 5 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) = U )
13	7	simp3bi 1038	. . . . 5 |- ( U e. ( 0 (,) _e ) -> U < _e )
14	12, 13	eqbrtrd 4104	. . . 4 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) < _e )
15		1red 8157	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 1 e. RR )
16		ltmuldiv 9017	. . . . 5 |- ( ( 1 e. RR /\ _e e. RR /\ ( U e. RR /\ 0 < U ) ) -> ( ( 1 x. U ) < _e <-> 1 < ( _e / U ) ) )
17	15, 2, 3, 8, 16	syl112anc 1275	. . . 4 |- ( U e. ( 0 (,) _e ) -> ( ( 1 x. U ) < _e <-> 1 < ( _e / U ) ) )
18	14, 17	mpbid 147	. . 3 |- ( U e. ( 0 (,) _e ) -> 1 < ( _e / U ) )
19		reeff1olem 15439	. . 3 |- ( ( ( _e / U ) e. RR /\ 1 < ( _e / U ) ) -> E. y e. RR ( exp ` y ) = ( _e / U ) )
20	10, 18, 19	syl2anc 411	. 2 |- ( U e. ( 0 (,) _e ) -> E. y e. RR ( exp ` y ) = ( _e / U ) )
21		1red 8157	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> 1 e. RR )
22		simprl 529	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> y e. RR )
23	21, 22	resubcld 8523	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( 1 - y ) e. RR )
24		1cnd 8158	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> 1 e. CC )
25	22	recnd 8171	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> y e. CC )
26		efsub 12187	. . . . 5 |- ( ( 1 e. CC /\ y e. CC ) -> ( exp ` ( 1 - y ) ) = ( ( exp ` 1 ) / ( exp ` y ) ) )
27	24, 25, 26	syl2anc 411	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` ( 1 - y ) ) = ( ( exp ` 1 ) / ( exp ` y ) ) )
28		simprr 531	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) = ( _e / U ) )
29		df-e 12155	. . . . . . . 8 |- _e = ( exp ` 1 )
30	29	oveq1i 6010	. . . . . . 7 |- ( _e / U ) = ( ( exp ` 1 ) / U )
31	28, 30	eqtr2di 2279	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( exp ` 1 ) / U ) = ( exp ` y ) )
32		efcl 12170	. . . . . . . 8 |- ( 1 e. CC -> ( exp ` 1 ) e. CC )
33	24, 32	syl 14	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` 1 ) e. CC )
34		efcl 12170	. . . . . . . 8 |- ( y e. CC -> ( exp ` y ) e. CC )
35	25, 34	syl 14	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) e. CC )
36	11	adantr 276	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> U e. CC )
37	9	adantr 276	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> U # 0 )
38	33, 35, 36, 37	divmulap2d 8967	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( ( exp ` 1 ) / U ) = ( exp ` y ) <-> ( exp ` 1 ) = ( U x. ( exp ` y ) ) ) )
39	31, 38	mpbid 147	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` 1 ) = ( U x. ( exp ` y ) ) )
40	22	rpefcld 12192	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) e. RR+ )
41	40	rpap0d 9894	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) # 0 )
42	33, 36, 35, 41	divmulap3d 8968	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( ( exp ` 1 ) / ( exp ` y ) ) = U <-> ( exp ` 1 ) = ( U x. ( exp ` y ) ) ) )
43	39, 42	mpbird 167	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( exp ` 1 ) / ( exp ` y ) ) = U )
44	27, 43	eqtrd 2262	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` ( 1 - y ) ) = U )
45		fveqeq2 5635	. . . 4 |- ( x = ( 1 - y ) -> ( ( exp ` x ) = U <-> ( exp ` ( 1 - y ) ) = U ) )
46	45	rspcev 2907	. . 3 |- ( ( ( 1 - y ) e. RR /\ ( exp ` ( 1 - y ) ) = U ) -> E. x e. RR ( exp ` x ) = U )
47	23, 44, 46	syl2anc 411	. 2 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> E. x e. RR ( exp ` x ) = U )
48	20, 47	rexlimddv 2653	1 |- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   x. cmul 8000   RR*cxr 8176   < clt 8177   - cmin 8313   # cap 8724   / cdiv 8815   (,)cioo 10080   expce 12148   _eceu 12149

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115  ax-pre-suploc 8116  ax-addf 8117  ax-mulf 8118

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-of 6216  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-map 6795  df-pm 6796  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-xneg 9964  df-xadd 9965  df-ioo 10084  df-ico 10086  df-icc 10087  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-bc 10965  df-ihash 10993  df-shft 11321  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860  df-ef 12154  df-e 12155  df-rest 13269  df-topgen 13288  df-psmet 14501  df-xmet 14502  df-met 14503  df-bl 14504  df-mopn 14505  df-top 14666  df-topon 14679  df-bases 14711  df-ntr 14764  df-cn 14856  df-cnp 14857  df-tx 14921  df-cncf 15239  df-limced 15324  df-dvap 15325

This theorem is referenced by:  reeff1o  15441