Theorem pntlemc 22970

Description: Lemma for pnt 22989. 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 22969	. . . . 5 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
10	9	simp2d 1001	. . . 4 |- ( ph -> D e. RR+ )
11	2, 10	rpdivcld 11148	. . 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 11148	. . . . 5 |- ( ph -> ( B / E ) e. RR+ )
15	14	rpred 11131	. . . 4 |- ( ph -> ( B / E ) e. RR )
16	15	rpefcld 13500	. . 3 |- ( ph -> ( exp ` ( B / E ) ) e. RR+ )
17	13, 16	syl5eqel 2543	. 2 |- ( ph -> K e. RR+ )
18	12	rpred 11131	. . . 4 |- ( ph -> E e. RR )
19	12	rpgt0d 11134	. . . 4 |- ( ph -> 0 < E )
20	2	rpred 11131	. . . . . . . 8 |- ( ph -> U e. RR )
21	4	rpred 11131	. . . . . . . 8 |- ( ph -> A e. RR )
22	10	rpred 11131	. . . . . . . 8 |- ( ph -> D e. RR )
23		pntlem1.u2	. . . . . . . 8 |- ( ph -> U <_ A )
24	21	ltp1d 10367	. . . . . . . . 9 |- ( ph -> A < ( A + 1 ) )
25	24, 7	syl6breqr 4433	. . . . . . . 8 |- ( ph -> A < D )
26	20, 21, 22, 23, 25	lelttrd 9633	. . . . . . 7 |- ( ph -> U < D )
27	10	rpcnd 11133	. . . . . . . 8 |- ( ph -> D e. CC )
28	27	mulid1d 9507	. . . . . . 7 |- ( ph -> ( D x. 1 ) = D )
29	26, 28	breqtrrd 4419	. . . . . 6 |- ( ph -> U < ( D x. 1 ) )
30		1red 9505	. . . . . . 7 |- ( ph -> 1 e. RR )
31	20, 30, 10	ltdivmuld 11178	. . . . . 6 |- ( ph -> ( ( U / D ) < 1 <-> U < ( D x. 1 ) ) )
32	29, 31	mpbird 232	. . . . 5 |- ( ph -> ( U / D ) < 1 )
33	1, 32	syl5eqbr 4426	. . . 4 |- ( ph -> E < 1 )
34		0xr 9534	. . . . 5 |- 0 e. RR*
35		1re 9489	. . . . . 6 |- 1 e. RR
36	35	rexri 9540	. . . . 5 |- 1 e. RR*
37		elioo2 11445	. . . . 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 13511	. . . . 5 |- ( ( B / E ) e. RR+ -> 1 < ( exp ` ( B / E ) ) )
41	14, 40	syl 16	. . . 4 |- ( ph -> 1 < ( exp ` ( B / E ) ) )
42	41, 13	syl6breqr 4433	. . 3 |- ( ph -> 1 < K )
43		ltaddrp 11127	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
44	35, 4, 43	sylancr 663	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
45	2	rpcnne0d 11140	. . . . . . . 8 |- ( ph -> ( U e. CC /\ U =/= 0 ) )
46		divid 10125	. . . . . . . 8 |- ( ( U e. CC /\ U =/= 0 ) -> ( U / U ) = 1 )
47	45, 46	syl 16	. . . . . . 7 |- ( ph -> ( U / U ) = 1 )
48	4	rpcnd 11133	. . . . . . . . 9 |- ( ph -> A e. CC )
49		ax-1cn 9444	. . . . . . . . 9 |- 1 e. CC
50		addcom 9659	. . . . . . . . 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 2504	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
53	44, 47, 52	3brtr4d 4423	. . . . . 6 |- ( ph -> ( U / U ) < D )
54	20, 2, 10, 53	ltdiv23d 11187	. . . . 5 |- ( ph -> ( U / D ) < U )
55	1, 54	syl5eqbr 4426	. . . 4 |- ( ph -> E < U )
56		difrp 11128	. . . . 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 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4393   |-> cmpt 4451   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   + caddc 9389   x. cmul 9391   RR*cxr 9521   < clt 9522   <_ cle 9523   - cmin 9699   / cdiv 10097   2c2 10475   3c3 10476  ;cdc 10859   RR+crp 11095   (,)cioo 11404   ^cexp 11975   expce 13458  ψcchp 22556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-ioo 11408  df-ico 11410  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464

This theorem is referenced by:  pntlema  22971  pntlemb  22972  pntlemg  22973  pntlemh  22974  pntlemq  22976  pntlemr  22977  pntlemj  22978  pntlemi  22979  pntlemf  22980  pntlemo  22982  pntleme  22983  pntlemp  22985