Theorem pntlemc 23756

Description: Lemma for pnt 23775. 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 23755	. . . . 5 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
10	9	simp2d 1010	. . . 4 |- ( ph -> D e. RR+ )
11	2, 10	rpdivcld 11283	. . 3 |- ( ph -> ( U / D ) e. RR+ )
12	1, 11	syl5eqel 2535	. 2 |- ( ph -> E e. RR+ )
13		pntlem1.k	. . 3 |- K = ( exp ` ( B / E ) )
14	5, 12	rpdivcld 11283	. . . . 5 |- ( ph -> ( B / E ) e. RR+ )
15	14	rpred 11266	. . . 4 |- ( ph -> ( B / E ) e. RR )
16	15	rpefcld 13821	. . 3 |- ( ph -> ( exp ` ( B / E ) ) e. RR+ )
17	13, 16	syl5eqel 2535	. 2 |- ( ph -> K e. RR+ )
18	12	rpred 11266	. . . 4 |- ( ph -> E e. RR )
19	12	rpgt0d 11269	. . . 4 |- ( ph -> 0 < E )
20	2	rpred 11266	. . . . . . . 8 |- ( ph -> U e. RR )
21	4	rpred 11266	. . . . . . . 8 |- ( ph -> A e. RR )
22	10	rpred 11266	. . . . . . . 8 |- ( ph -> D e. RR )
23		pntlem1.u2	. . . . . . . 8 |- ( ph -> U <_ A )
24	21	ltp1d 10483	. . . . . . . . 9 |- ( ph -> A < ( A + 1 ) )
25	24, 7	syl6breqr 4477	. . . . . . . 8 |- ( ph -> A < D )
26	20, 21, 22, 23, 25	lelttrd 9743	. . . . . . 7 |- ( ph -> U < D )
27	10	rpcnd 11268	. . . . . . . 8 |- ( ph -> D e. CC )
28	27	mulid1d 9616	. . . . . . 7 |- ( ph -> ( D x. 1 ) = D )
29	26, 28	breqtrrd 4463	. . . . . 6 |- ( ph -> U < ( D x. 1 ) )
30		1red 9614	. . . . . . 7 |- ( ph -> 1 e. RR )
31	20, 30, 10	ltdivmuld 11313	. . . . . 6 |- ( ph -> ( ( U / D ) < 1 <-> U < ( D x. 1 ) ) )
32	29, 31	mpbird 232	. . . . 5 |- ( ph -> ( U / D ) < 1 )
33	1, 32	syl5eqbr 4470	. . . 4 |- ( ph -> E < 1 )
34		0xr 9643	. . . . 5 |- 0 e. RR*
35		1re 9598	. . . . . 6 |- 1 e. RR
36	35	rexri 9649	. . . . 5 |- 1 e. RR*
37		elioo2 11580	. . . . 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 1181	. . 3 |- ( ph -> E e. ( 0 (,) 1 ) )
40		efgt1 13832	. . . . 5 |- ( ( B / E ) e. RR+ -> 1 < ( exp ` ( B / E ) ) )
41	14, 40	syl 16	. . . 4 |- ( ph -> 1 < ( exp ` ( B / E ) ) )
42	41, 13	syl6breqr 4477	. . 3 |- ( ph -> 1 < K )
43		ltaddrp 11262	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
44	35, 4, 43	sylancr 663	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
45	2	rpcnne0d 11275	. . . . . . . 8 |- ( ph -> ( U e. CC /\ U =/= 0 ) )
46		divid 10241	. . . . . . . 8 |- ( ( U e. CC /\ U =/= 0 ) -> ( U / U ) = 1 )
47	45, 46	syl 16	. . . . . . 7 |- ( ph -> ( U / U ) = 1 )
48	4	rpcnd 11268	. . . . . . . . 9 |- ( ph -> A e. CC )
49		ax-1cn 9553	. . . . . . . . 9 |- 1 e. CC
50		addcom 9769	. . . . . . . . 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 2496	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
53	44, 47, 52	3brtr4d 4467	. . . . . 6 |- ( ph -> ( U / U ) < D )
54	20, 2, 10, 53	ltdiv23d 11322	. . . . 5 |- ( ph -> ( U / D ) < U )
55	1, 54	syl5eqbr 4470	. . . 4 |- ( ph -> E < U )
56		difrp 11263	. . . . 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 1177	. 2 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
60	12, 17, 59	3jca 1177	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 974   = wceq 1383   e. wcel 1804   =/= wne 2638   class class class wbr 4437   |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   + caddc 9498   x. cmul 9500   RR*cxr 9630   < clt 9631   <_ cle 9632   - cmin 9810   / cdiv 10213   2c2 10592   3c3 10593  ;cdc 10985   RR+crp 11230   (,)cioo 11539   ^cexp 12147   expce 13778  ψcchp 23342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-rp 11231  df-ioo 11543  df-ico 11545  df-fz 11683  df-fzo 11806  df-fl 11910  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-shft 12881  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-limsup 13275  df-clim 13292  df-rlim 13293  df-sum 13490  df-ef 13784

This theorem is referenced by:  pntlema  23757  pntlemb  23758  pntlemg  23759  pntlemh  23760  pntlemq  23762  pntlemr  23763  pntlemj  23764  pntlemi  23765  pntlemf  23766  pntlemo  23768  pntleme  23769  pntlemp  23771