Theorem pntlemc 22819

Description: Lemma for pnt 22838. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, U is α, E is ε, and K is K. (Contributed by Mario Carneiro, 13-Apr-2016.)

Hypotheses

Ref	Expression
pntlem1.r	|- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
pntlem1.a	|- ( ph -> A e. RR+ )
pntlem1.b	|- ( ph -> B e. RR+ )
pntlem1.l	|- ( ph -> L e. ( 0 (,) 1 ) )
pntlem1.d	|- D = ( A + 1 )
pntlem1.f	|- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
pntlem1.u	|- ( ph -> U e. RR+ )
pntlem1.u2	|- ( ph -> U <_ A )
pntlem1.e	|- E = ( U / D )
pntlem1.k	|- K = ( exp ` ( B / E ) )

Assertion

Ref	Expression
pntlemc	|- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )

Distinct variable group:   E, a

Allowed substitution hints:   ph( a)   A( a)   B( a)   D( a)   R( a)   U( a)   F( a)   K( a)   L( a)

Proof of Theorem pntlemc

Step	Hyp	Ref	Expression
1		pntlem1.e	. . 3 |- E = ( U / D )
2		pntlem1.u	. . . 4 |- ( ph -> U e. RR+ )
3		pntlem1.r	. . . . . 6 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
4		pntlem1.a	. . . . . 6 |- ( ph -> A e. RR+ )
5		pntlem1.b	. . . . . 6 |- ( ph -> B e. RR+ )
6		pntlem1.l	. . . . . 6 |- ( ph -> L e. ( 0 (,) 1 ) )
7		pntlem1.d	. . . . . 6 |- D = ( A + 1 )
8		pntlem1.f	. . . . . 6 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
9	3, 4, 5, 6, 7, 8	pntlemd 22818	. . . . 5 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
10	9	simp2d 1001	. . . 4 |- ( ph -> D e. RR+ )
11	2, 10	rpdivcld 11036	. . 3 |- ( ph -> ( U / D ) e. RR+ )
12	1, 11	syl5eqel 2522	. 2 |- ( ph -> E e. RR+ )
13		pntlem1.k	. . 3 |- K = ( exp ` ( B / E ) )
14	5, 12	rpdivcld 11036	. . . . 5 |- ( ph -> ( B / E ) e. RR+ )
15	14	rpred 11019	. . . 4 |- ( ph -> ( B / E ) e. RR )
16	15	rpefcld 13381	. . 3 |- ( ph -> ( exp ` ( B / E ) ) e. RR+ )
17	13, 16	syl5eqel 2522	. 2 |- ( ph -> K e. RR+ )
18	12	rpred 11019	. . . 4 |- ( ph -> E e. RR )
19	12	rpgt0d 11022	. . . 4 |- ( ph -> 0 < E )
20	2	rpred 11019	. . . . . . . 8 |- ( ph -> U e. RR )
21	4	rpred 11019	. . . . . . . 8 |- ( ph -> A e. RR )
22	10	rpred 11019	. . . . . . . 8 |- ( ph -> D e. RR )
23		pntlem1.u2	. . . . . . . 8 |- ( ph -> U <_ A )
24	21	ltp1d 10255	. . . . . . . . 9 |- ( ph -> A < ( A + 1 ) )
25	24, 7	syl6breqr 4327	. . . . . . . 8 |- ( ph -> A < D )
26	20, 21, 22, 23, 25	lelttrd 9521	. . . . . . 7 |- ( ph -> U < D )
27	10	rpcnd 11021	. . . . . . . 8 |- ( ph -> D e. CC )
28	27	mulid1d 9395	. . . . . . 7 |- ( ph -> ( D x. 1 ) = D )
29	26, 28	breqtrrd 4313	. . . . . 6 |- ( ph -> U < ( D x. 1 ) )
30		1red 9393	. . . . . . 7 |- ( ph -> 1 e. RR )
31	20, 30, 10	ltdivmuld 11066	. . . . . 6 |- ( ph -> ( ( U / D ) < 1 <-> U < ( D x. 1 ) ) )
32	29, 31	mpbird 232	. . . . 5 |- ( ph -> ( U / D ) < 1 )
33	1, 32	syl5eqbr 4320	. . . 4 |- ( ph -> E < 1 )
34		0xr 9422	. . . . 5 |- 0 e. RR*
35		1re 9377	. . . . . 6 |- 1 e. RR
36	35	rexri 9428	. . . . 5 |- 1 e. RR*
37		elioo2 11333	. . . . 5 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) ) )
38	34, 36, 37	mp2an 672	. . . 4 |- ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) )
39	18, 19, 33, 38	syl3anbrc 1172	. . 3 |- ( ph -> E e. ( 0 (,) 1 ) )
40		efgt1 13392	. . . . 5 |- ( ( B / E ) e. RR+ -> 1 < ( exp ` ( B / E ) ) )
41	14, 40	syl 16	. . . 4 |- ( ph -> 1 < ( exp ` ( B / E ) ) )
42	41, 13	syl6breqr 4327	. . 3 |- ( ph -> 1 < K )
43		ltaddrp 11015	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
44	35, 4, 43	sylancr 663	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
45	2	rpcnne0d 11028	. . . . . . . 8 |- ( ph -> ( U e. CC /\ U =/= 0 ) )
46		divid 10013	. . . . . . . 8 |- ( ( U e. CC /\ U =/= 0 ) -> ( U / U ) = 1 )
47	45, 46	syl 16	. . . . . . 7 |- ( ph -> ( U / U ) = 1 )
48	4	rpcnd 11021	. . . . . . . . 9 |- ( ph -> A e. CC )
49		ax-1cn 9332	. . . . . . . . 9 |- 1 e. CC
50		addcom 9547	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
51	48, 49, 50	sylancl 662	. . . . . . . 8 |- ( ph -> ( A + 1 ) = ( 1 + A ) )
52	7, 51	syl5eq 2482	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
53	44, 47, 52	3brtr4d 4317	. . . . . 6 |- ( ph -> ( U / U ) < D )
54	20, 2, 10, 53	ltdiv23d 11075	. . . . 5 |- ( ph -> ( U / D ) < U )
55	1, 54	syl5eqbr 4320	. . . 4 |- ( ph -> E < U )
56		difrp 11016	. . . . 5 |- ( ( E e. RR /\ U e. RR ) -> ( E < U <-> ( U - E ) e. RR+ ) )
57	18, 20, 56	syl2anc 661	. . . 4 |- ( ph -> ( E < U <-> ( U - E ) e. RR+ ) )
58	55, 57	mpbid 210	. . 3 |- ( ph -> ( U - E ) e. RR+ )
59	39, 42, 58	3jca 1168	. 2 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
60	12, 17, 59	3jca 1168	1 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   class class class wbr 4287   e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   + caddc 9277   x. cmul 9279   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   / cdiv 9985   2c2 10363   3c3 10364  ;cdc 10747   RR+crp 10983   (,)cioo 11292   ^cexp 11857   expce 13339  ψcchp 22405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-rp 10984  df-ioo 11296  df-ico 11298  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345

This theorem is referenced by:  pntlema  22820  pntlemb  22821  pntlemg  22822  pntlemh  22823  pntlemq  22825  pntlemr  22826  pntlemj  22827  pntlemi  22828  pntlemf  22829  pntlemo  22831  pntleme  22832  pntlemp  22834