Theorem reeff1oleme 14645

Description: Lemma for reeff1o 14646. (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 11709	. . . . 5 |- _e e. RR
2	1	a1i 9	. . . 4 |- ( U e. ( 0 (,) _e ) -> _e e. RR )
3		elioore 9941	. . . 4 |- ( U e. ( 0 (,) _e ) -> U e. RR )
4		0xr 8033	. . . . . . 7 |- 0 e. RR*
5	1	rexri 8044	. . . . . . 7 |- _e e. RR*
6		elioo2 9950	. . . . . . 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 1015	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 0 < U )
9	3, 8	gt0ap0d 8615	. . . 4 |- ( U e. ( 0 (,) _e ) -> U # 0 )
10	2, 3, 9	redivclapd 8821	. . 3 |- ( U e. ( 0 (,) _e ) -> ( _e / U ) e. RR )
11	3	recnd 8015	. . . . . 6 |- ( U e. ( 0 (,) _e ) -> U e. CC )
12	11	mulid2d 8005	. . . . 5 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) = U )
13	7	simp3bi 1016	. . . . 5 |- ( U e. ( 0 (,) _e ) -> U < _e )
14	12, 13	eqbrtrd 4040	. . . 4 |- ( U e. ( 0 (,) _e ) -> ( 1 x. U ) < _e )
15		1red 8001	. . . . 5 |- ( U e. ( 0 (,) _e ) -> 1 e. RR )
16		ltmuldiv 8860	. . . . 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 1253	. . . 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 14644	. . 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 8001	. . . 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 8367	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( 1 - y ) e. RR )
24		1cnd 8002	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> 1 e. CC )
25	22	recnd 8015	. . . . 5 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> y e. CC )
26		efsub 11720	. . . . 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 11688	. . . . . . . 8 |- _e = ( exp ` 1 )
30	29	oveq1i 5905	. . . . . . 7 |- ( _e / U ) = ( ( exp ` 1 ) / U )
31	28, 30	eqtr2di 2239	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( ( exp ` 1 ) / U ) = ( exp ` y ) )
32		efcl 11703	. . . . . . . 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 11703	. . . . . . . 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 8810	. . . . . 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 11725	. . . . . . 7 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) e. RR+ )
41	40	rpap0d 9731	. . . . . 6 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` y ) # 0 )
42	33, 36, 35, 41	divmulap3d 8811	. . . . 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 2222	. . 3 |- ( ( U e. ( 0 (,) _e ) /\ ( y e. RR /\ ( exp ` y ) = ( _e / U ) ) ) -> ( exp ` ( 1 - y ) ) = U )
45		fveqeq2 5543	. . . 4 |- ( x = ( 1 - y ) -> ( ( exp ` x ) = U <-> ( exp ` ( 1 - y ) ) = U ) )
46	45	rspcev 2856	. . 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 2612	1 |- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   E.wrex 2469   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   1c1 7841   x. cmul 7845   RR*cxr 8020   < clt 8021   - cmin 8157   # cap 8567   / cdiv 8658   (,)cioo 9917   expce 11681   _eceu 11682

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960  ax-pre-suploc 7961  ax-addf 7962  ax-mulf 7963

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-of 6105  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-oadd 6444  df-er 6558  df-map 6675  df-pm 6676  df-en 6766  df-dom 6767  df-fin 6768  df-sup 7012  df-inf 7013  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-xneg 9801  df-xadd 9802  df-ioo 9921  df-ico 9923  df-icc 9924  df-fz 10038  df-fzo 10172  df-seqfrec 10476  df-exp 10550  df-fac 10737  df-bc 10759  df-ihash 10787  df-shft 10855  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-clim 11318  df-sumdc 11393  df-ef 11687  df-e 11688  df-rest 12743  df-topgen 12762  df-psmet 13853  df-xmet 13854  df-met 13855  df-bl 13856  df-mopn 13857  df-top 13950  df-topon 13963  df-bases 13995  df-ntr 14048  df-cn 14140  df-cnp 14141  df-tx 14205  df-cncf 14510  df-limced 14577  df-dvap 14578

This theorem is referenced by:  reeff1o  14646