Theorem pntlema 25285

Description: Lemma for pnt 25303. Closure for the constants used in the proof. The mammoth expression W is a number large enough to satisfy all the lower bounds needed for Z. For comparison with Equation 10.6.27 of [Shapiro], p. 434, Y is x₂, X is x₁, C is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and W is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (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 ) )
pntlem1.y	|- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
pntlem1.x	|- ( ph -> ( X e. RR+ /\ Y < X ) )
pntlem1.c	|- ( ph -> C e. RR+ )
pntlem1.w	|- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )

Assertion

Ref	Expression
pntlema	|- ( ph -> W e. RR+ )

Distinct variable group:   E, a

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

Proof of Theorem pntlema

Step	Hyp	Ref	Expression
1		pntlem1.w	. 2 |- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
2		pntlem1.y	. . . . . 6 |- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
3	2	simpld 475	. . . . 5 |- ( ph -> Y e. RR+ )
4		4nn 11187	. . . . . . 7 |- 4 e. NN
5		nnrp 11842	. . . . . . 7 |- ( 4 e. NN -> 4 e. RR+ )
6	4, 5	ax-mp 5	. . . . . 6 |- 4 e. RR+
7		pntlem1.r	. . . . . . . . 9 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
8		pntlem1.a	. . . . . . . . 9 |- ( ph -> A e. RR+ )
9		pntlem1.b	. . . . . . . . 9 |- ( ph -> B e. RR+ )
10		pntlem1.l	. . . . . . . . 9 |- ( ph -> L e. ( 0 (,) 1 ) )
11		pntlem1.d	. . . . . . . . 9 |- D = ( A + 1 )
12		pntlem1.f	. . . . . . . . 9 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
13	7, 8, 9, 10, 11, 12	pntlemd 25283	. . . . . . . 8 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
14	13	simp1d 1073	. . . . . . 7 |- ( ph -> L e. RR+ )
15		pntlem1.u	. . . . . . . . 9 |- ( ph -> U e. RR+ )
16		pntlem1.u2	. . . . . . . . 9 |- ( ph -> U <_ A )
17		pntlem1.e	. . . . . . . . 9 |- E = ( U / D )
18		pntlem1.k	. . . . . . . . 9 |- K = ( exp ` ( B / E ) )
19	7, 8, 9, 10, 11, 12, 15, 16, 17, 18	pntlemc 25284	. . . . . . . 8 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
20	19	simp1d 1073	. . . . . . 7 |- ( ph -> E e. RR+ )
21	14, 20	rpmulcld 11888	. . . . . 6 |- ( ph -> ( L x. E ) e. RR+ )
22		rpdivcl 11856	. . . . . 6 |- ( ( 4 e. RR+ /\ ( L x. E ) e. RR+ ) -> ( 4 / ( L x. E ) ) e. RR+ )
23	6, 21, 22	sylancr 695	. . . . 5 |- ( ph -> ( 4 / ( L x. E ) ) e. RR+ )
24	3, 23	rpaddcld 11887	. . . 4 |- ( ph -> ( Y + ( 4 / ( L x. E ) ) ) e. RR+ )
25		2z 11409	. . . 4 |- 2 e. ZZ
26		rpexpcl 12879	. . . 4 |- ( ( ( Y + ( 4 / ( L x. E ) ) ) e. RR+ /\ 2 e. ZZ ) -> ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) e. RR+ )
27	24, 25, 26	sylancl 694	. . 3 |- ( ph -> ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) e. RR+ )
28		pntlem1.x	. . . . . . 7 |- ( ph -> ( X e. RR+ /\ Y < X ) )
29	28	simpld 475	. . . . . 6 |- ( ph -> X e. RR+ )
30	19	simp2d 1074	. . . . . . 7 |- ( ph -> K e. RR+ )
31		rpexpcl 12879	. . . . . . 7 |- ( ( K e. RR+ /\ 2 e. ZZ ) -> ( K ^ 2 ) e. RR+ )
32	30, 25, 31	sylancl 694	. . . . . 6 |- ( ph -> ( K ^ 2 ) e. RR+ )
33	29, 32	rpmulcld 11888	. . . . 5 |- ( ph -> ( X x. ( K ^ 2 ) ) e. RR+ )
34		4z 11411	. . . . 5 |- 4 e. ZZ
35		rpexpcl 12879	. . . . 5 |- ( ( ( X x. ( K ^ 2 ) ) e. RR+ /\ 4 e. ZZ ) -> ( ( X x. ( K ^ 2 ) ) ^ 4 ) e. RR+ )
36	33, 34, 35	sylancl 694	. . . 4 |- ( ph -> ( ( X x. ( K ^ 2 ) ) ^ 4 ) e. RR+ )
37		3nn0 11310	. . . . . . . . . . 11 |- 3 e. NN0
38		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
39	37, 38	decnncl 11518	. . . . . . . . . 10 |- ; 3 2 e. NN
40		nnrp 11842	. . . . . . . . . 10 |- (; 3 2 e. NN -> ; 3 2 e. RR+ )
41	39, 40	ax-mp 5	. . . . . . . . 9 |- ; 3 2 e. RR+
42		rpmulcl 11855	. . . . . . . . 9 |- ( (; 3 2 e. RR+ /\ B e. RR+ ) -> (; 3 2 x. B ) e. RR+ )
43	41, 9, 42	sylancr 695	. . . . . . . 8 |- ( ph -> (; 3 2 x. B ) e. RR+ )
44	19	simp3d 1075	. . . . . . . . . 10 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
45	44	simp3d 1075	. . . . . . . . 9 |- ( ph -> ( U - E ) e. RR+ )
46		rpexpcl 12879	. . . . . . . . . . 11 |- ( ( E e. RR+ /\ 2 e. ZZ ) -> ( E ^ 2 ) e. RR+ )
47	20, 25, 46	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( E ^ 2 ) e. RR+ )
48	14, 47	rpmulcld 11888	. . . . . . . . 9 |- ( ph -> ( L x. ( E ^ 2 ) ) e. RR+ )
49	45, 48	rpmulcld 11888	. . . . . . . 8 |- ( ph -> ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) e. RR+ )
50	43, 49	rpdivcld 11889	. . . . . . 7 |- ( ph -> ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) e. RR+ )
51		3nn 11186	. . . . . . . . . 10 |- 3 e. NN
52		nnrp 11842	. . . . . . . . . 10 |- ( 3 e. NN -> 3 e. RR+ )
53	51, 52	ax-mp 5	. . . . . . . . 9 |- 3 e. RR+
54		rpmulcl 11855	. . . . . . . . 9 |- ( ( U e. RR+ /\ 3 e. RR+ ) -> ( U x. 3 ) e. RR+ )
55	15, 53, 54	sylancl 694	. . . . . . . 8 |- ( ph -> ( U x. 3 ) e. RR+ )
56		pntlem1.c	. . . . . . . 8 |- ( ph -> C e. RR+ )
57	55, 56	rpaddcld 11887	. . . . . . 7 |- ( ph -> ( ( U x. 3 ) + C ) e. RR+ )
58	50, 57	rpmulcld 11888	. . . . . 6 |- ( ph -> ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) e. RR+ )
59	58	rpred 11872	. . . . 5 |- ( ph -> ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) e. RR )
60	59	rpefcld 14835	. . . 4 |- ( ph -> ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) e. RR+ )
61	36, 60	rpaddcld 11887	. . 3 |- ( ph -> ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) e. RR+ )
62	27, 61	rpaddcld 11887	. 2 |- ( ph -> ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) ) e. RR+ )
63	1, 62	syl5eqel 2705	1 |- ( ph -> W e. RR+ )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   ZZcz 11377  ;cdc 11493   RR+crp 11832   (,)cioo 12175   ^cexp 12860   expce 14792  ψcchp 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ioo 12179  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798

This theorem is referenced by:  pntlemb  25286  pntleme  25297