Theorem reeff1oleme 15502

Description: Lemma for reeff1o 15503. (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 12236	. . . . 5 |- _e e. RR
2	1	a1i 9	. . . 4 |- ( U e. ( 0 (,) _e ) -> _e e. RR )
3		elioore 10147	. . . 4 |- ( U e. ( 0 (,) _e ) -> U e. RR )
4		0xr 8226	. . . . . . 7 |- 0 e. RR*
5	1	rexri 8237	. . . . . . 7 |- _e e. RR*
6		elioo2 10156	. . . . . . 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 1039	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 0 < U )
9	3, 8	gt0ap0d 8809	. . . 4 |- ( U e. ( 0 (,) _e ) -> U # 0 )
10	2, 3, 9	redivclapd 9015	. . 3 |- ( U e. ( 0 (,) _e ) -> ( _e / U ) e. RR )
11	3	recnd 8208	. . . . . 6 |- ( U e. ( 0 (,) _e ) -> U e. CC )
12	11	mulid2d 8198	. . . . 5 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) = U )
13	7	simp3bi 1040	. . . . 5 |- ( U e. ( 0 (,) _e ) -> U < _e )
14	12, 13	eqbrtrd 4110	. . . 4 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) < _e )
15		1red 8194	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 1 e. RR )
16		ltmuldiv 9054	. . . . 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 1277	. . . 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 15501	. . 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 8194	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> 1 e. RR )
22		simprl 531	. . . 4 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> y e. RR )
23	21, 22	resubcld 8560	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( 1 - y ) e. RR )
24		1cnd 8195	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> 1 e. CC )
25	22	recnd 8208	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> y e. CC )
26		efsub 12247	. . . . 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 533	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) = ( _e / U ) )
29		df-e 12215	. . . . . . . 8 |- _e = ( exp ` 1 )
30	29	oveq1i 6028	. . . . . . 7 |- ( _e / U ) = ( ( exp ` 1 ) / U )
31	28, 30	eqtr2di 2281	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( exp ` 1 ) / U ) = ( exp ` y ) )
32		efcl 12230	. . . . . . . 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 12230	. . . . . . . 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 9004	. . . . . 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 12252	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) e. RR+ )
41	40	rpap0d 9937	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) # 0 )
42	33, 36, 35, 41	divmulap3d 9005	. . . . 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 2264	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` ( 1 - y ) ) = U )
45		fveqeq2 5648	. . . 4 |- ( x = ( 1 - y ) -> ( ( exp ` x ) = U <-> ( exp ` ( 1 - y ) ) = U ) )
46	45	rspcev 2910	. . 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 2655	1 |- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   x. cmul 8037   RR*cxr 8213   < clt 8214   - cmin 8350   # cap 8761   / cdiv 8852   (,)cioo 10123   expce 12208   _eceu 12209

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-pre-suploc 8153  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-map 6819  df-pm 6820  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-ico 10129  df-icc 10130  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-bc 11011  df-ihash 11039  df-shft 11380  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-clim 11844  df-sumdc 11919  df-ef 12214  df-e 12215  df-rest 13329  df-topgen 13348  df-psmet 14563  df-xmet 14564  df-met 14565  df-bl 14566  df-mopn 14567  df-top 14728  df-topon 14741  df-bases 14773  df-ntr 14826  df-cn 14918  df-cnp 14919  df-tx 14983  df-cncf 15301  df-limced 15386  df-dvap 15387

This theorem is referenced by:  reeff1o  15503