Theorem pntlemc 24481

Description: Lemma for pnt 24500. 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 24480	. . . . 5 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
10	9	simp2d 1027	. . . 4 |- ( ph -> D e. RR+ )
11	2, 10	rpdivcld 11386	. . 3 |- ( ph -> ( U / D ) e. RR+ )
12	1, 11	syl5eqel 2543	. 2 |- ( ph -> E e. RR+ )
13		pntlem1.k	. . 3 |- K = ( exp ` ( B / E ) )
14	5, 12	rpdivcld 11386	. . . . 5 |- ( ph -> ( B / E ) e. RR+ )
15	14	rpred 11369	. . . 4 |- ( ph -> ( B / E ) e. RR )
16	15	rpefcld 14207	. . 3 |- ( ph -> ( exp ` ( B / E ) ) e. RR+ )
17	13, 16	syl5eqel 2543	. 2 |- ( ph -> K e. RR+ )
18	12	rpred 11369	. . . 4 |- ( ph -> E e. RR )
19	12	rpgt0d 11372	. . . 4 |- ( ph -> 0 < E )
20	2	rpred 11369	. . . . . . . 8 |- ( ph -> U e. RR )
21	4	rpred 11369	. . . . . . . 8 |- ( ph -> A e. RR )
22	10	rpred 11369	. . . . . . . 8 |- ( ph -> D e. RR )
23		pntlem1.u2	. . . . . . . 8 |- ( ph -> U <_ A )
24	21	ltp1d 10564	. . . . . . . . 9 |- ( ph -> A < ( A + 1 ) )
25	24, 7	syl6breqr 4456	. . . . . . . 8 |- ( ph -> A < D )
26	20, 21, 22, 23, 25	lelttrd 9818	. . . . . . 7 |- ( ph -> U < D )
27	10	rpcnd 11371	. . . . . . . 8 |- ( ph -> D e. CC )
28	27	mulid1d 9685	. . . . . . 7 |- ( ph -> ( D x. 1 ) = D )
29	26, 28	breqtrrd 4442	. . . . . 6 |- ( ph -> U < ( D x. 1 ) )
30		1red 9683	. . . . . . 7 |- ( ph -> 1 e. RR )
31	20, 30, 10	ltdivmuld 11417	. . . . . 6 |- ( ph -> ( ( U / D ) < 1 <-> U < ( D x. 1 ) ) )
32	29, 31	mpbird 240	. . . . 5 |- ( ph -> ( U / D ) < 1 )
33	1, 32	syl5eqbr 4449	. . . 4 |- ( ph -> E < 1 )
34		0xr 9712	. . . . 5 |- 0 e. RR*
35		1re 9667	. . . . . 6 |- 1 e. RR
36	35	rexri 9718	. . . . 5 |- 1 e. RR*
37		elioo2 11705	. . . . 5 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) ) )
38	34, 36, 37	mp2an 683	. . . 4 |- ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) )
39	18, 19, 33, 38	syl3anbrc 1198	. . 3 |- ( ph -> E e. ( 0 (,) 1 ) )
40		efgt1 14218	. . . . 5 |- ( ( B / E ) e. RR+ -> 1 < ( exp ` ( B / E ) ) )
41	14, 40	syl 17	. . . 4 |- ( ph -> 1 < ( exp ` ( B / E ) ) )
42	41, 13	syl6breqr 4456	. . 3 |- ( ph -> 1 < K )
43		ltaddrp 11364	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
44	35, 4, 43	sylancr 674	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
45	2	rpcnne0d 11378	. . . . . . . 8 |- ( ph -> ( U e. CC /\ U =/= 0 ) )
46		divid 10324	. . . . . . . 8 |- ( ( U e. CC /\ U =/= 0 ) -> ( U / U ) = 1 )
47	45, 46	syl 17	. . . . . . 7 |- ( ph -> ( U / U ) = 1 )
48	4	rpcnd 11371	. . . . . . . . 9 |- ( ph -> A e. CC )
49		ax-1cn 9622	. . . . . . . . 9 |- 1 e. CC
50		addcom 9844	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
51	48, 49, 50	sylancl 673	. . . . . . . 8 |- ( ph -> ( A + 1 ) = ( 1 + A ) )
52	7, 51	syl5eq 2507	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
53	44, 47, 52	3brtr4d 4446	. . . . . 6 |- ( ph -> ( U / U ) < D )
54	20, 2, 10, 53	ltdiv23d 11431	. . . . 5 |- ( ph -> ( U / D ) < U )
55	1, 54	syl5eqbr 4449	. . . 4 |- ( ph -> E < U )
56		difrp 11365	. . . . 5 |- ( ( E e. RR /\ U e. RR ) -> ( E < U <-> ( U - E ) e. RR+ ) )
57	18, 20, 56	syl2anc 671	. . . 4 |- ( ph -> ( E < U <-> ( U - E ) e. RR+ ) )
58	55, 57	mpbid 215	. . 3 |- ( ph -> ( U - E ) e. RR+ )
59	39, 42, 58	3jca 1194	. 2 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
60	12, 17, 59	3jca 1194	1 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   class class class wbr 4415   |-> cmpt 4474   ` cfv 5600  (class class class)co 6314   CCcc 9562   RRcr 9563   0cc0 9564   1c1 9565   + caddc 9567   x. cmul 9569   RR*cxr 9699   < clt 9700   <_ cle 9701   - cmin 9885   / cdiv 10296   2c2 10686   3c3 10687  ;cdc 11079   RR+crp 11330   (,)cioo 11663   ^cexp 12303   expce 14162  ψcchp 24067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-rp 11331  df-ioo 11667  df-ico 11669  df-fz 11813  df-fzo 11946  df-fl 12059  df-seq 12245  df-exp 12304  df-fac 12491  df-bc 12519  df-hash 12547  df-shft 13178  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-limsup 13574  df-clim 13600  df-rlim 13601  df-sum 13801  df-ef 14169

This theorem is referenced by:  pntlema  24482  pntlemb  24483  pntlemg  24484  pntlemh  24485  pntlemq  24487  pntlemr  24488  pntlemj  24489  pntlemi  24490  pntlemf  24491  pntlemo  24493  pntleme  24494  pntlemp  24496