Theorem pntlemd 25283

Description: Lemma for pnt 25303. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, A is C^*, B is c₁, L is λ, D is c₂, and F is c₃. (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 ) ) )

Assertion

Ref	Expression
pntlemd	|- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )

Proof of Theorem pntlemd

Step	Hyp	Ref	Expression
1		ioossre 12235	. . . 4 |- ( 0 (,) 1 ) C_ RR
2		pntlem1.l	. . . 4 |- ( ph -> L e. ( 0 (,) 1 ) )
3	1, 2	sseldi 3601	. . 3 |- ( ph -> L e. RR )
4		eliooord 12233	. . . . 5 |- ( L e. ( 0 (,) 1 ) -> ( 0 < L /\ L < 1 ) )
5	2, 4	syl 17	. . . 4 |- ( ph -> ( 0 < L /\ L < 1 ) )
6	5	simpld 475	. . 3 |- ( ph -> 0 < L )
7	3, 6	elrpd 11869	. 2 |- ( ph -> L e. RR+ )
8		pntlem1.d	. . 3 |- D = ( A + 1 )
9		pntlem1.a	. . . 4 |- ( ph -> A e. RR+ )
10		1rp 11836	. . . 4 |- 1 e. RR+
11		rpaddcl 11854	. . . 4 |- ( ( A e. RR+ /\ 1 e. RR+ ) -> ( A + 1 ) e. RR+ )
12	9, 10, 11	sylancl 694	. . 3 |- ( ph -> ( A + 1 ) e. RR+ )
13	8, 12	syl5eqel 2705	. 2 |- ( ph -> D e. RR+ )
14		pntlem1.f	. . 3 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
15		1re 10039	. . . . . . . 8 |- 1 e. RR
16		ltaddrp 11867	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
17	15, 9, 16	sylancr 695	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
18	9	rpcnd 11874	. . . . . . . . 9 |- ( ph -> A e. CC )
19		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
20		addcom 10222	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
21	18, 19, 20	sylancl 694	. . . . . . . 8 |- ( ph -> ( A + 1 ) = ( 1 + A ) )
22	8, 21	syl5eq 2668	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
23	17, 22	breqtrrd 4681	. . . . . 6 |- ( ph -> 1 < D )
24	13	recgt1d 11886	. . . . . 6 |- ( ph -> ( 1 < D <-> ( 1 / D ) < 1 ) )
25	23, 24	mpbid 222	. . . . 5 |- ( ph -> ( 1 / D ) < 1 )
26	13	rprecred 11883	. . . . . 6 |- ( ph -> ( 1 / D ) e. RR )
27		difrp 11868	. . . . . 6 |- ( ( ( 1 / D ) e. RR /\ 1 e. RR ) -> ( ( 1 / D ) < 1 <-> ( 1 - ( 1 / D ) ) e. RR+ ) )
28	26, 15, 27	sylancl 694	. . . . 5 |- ( ph -> ( ( 1 / D ) < 1 <-> ( 1 - ( 1 / D ) ) e. RR+ ) )
29	25, 28	mpbid 222	. . . 4 |- ( ph -> ( 1 - ( 1 / D ) ) e. RR+ )
30		3nn0 11310	. . . . . . . . 9 |- 3 e. NN0
31		2nn 11185	. . . . . . . . 9 |- 2 e. NN
32	30, 31	decnncl 11518	. . . . . . . 8 |- ; 3 2 e. NN
33		nnrp 11842	. . . . . . . 8 |- (; 3 2 e. NN -> ; 3 2 e. RR+ )
34	32, 33	ax-mp 5	. . . . . . 7 |- ; 3 2 e. RR+
35		pntlem1.b	. . . . . . 7 |- ( ph -> B e. RR+ )
36		rpmulcl 11855	. . . . . . 7 |- ( (; 3 2 e. RR+ /\ B e. RR+ ) -> (; 3 2 x. B ) e. RR+ )
37	34, 35, 36	sylancr 695	. . . . . 6 |- ( ph -> (; 3 2 x. B ) e. RR+ )
38	7, 37	rpdivcld 11889	. . . . 5 |- ( ph -> ( L / (; 3 2 x. B ) ) e. RR+ )
39		2z 11409	. . . . . 6 |- 2 e. ZZ
40		rpexpcl 12879	. . . . . 6 |- ( ( D e. RR+ /\ 2 e. ZZ ) -> ( D ^ 2 ) e. RR+ )
41	13, 39, 40	sylancl 694	. . . . 5 |- ( ph -> ( D ^ 2 ) e. RR+ )
42	38, 41	rpdivcld 11889	. . . 4 |- ( ph -> ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) e. RR+ )
43	29, 42	rpmulcld 11888	. . 3 |- ( ph -> ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) ) e. RR+ )
44	14, 43	syl5eqel 2705	. 2 |- ( ph -> F e. RR+ )
45	7, 13, 44	3jca 1242	1 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377  ;cdc 11493   RR+crp 11832   (,)cioo 12175   ^cexp 12860  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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-seq 12802  df-exp 12861

This theorem is referenced by:  pntlemc  25284  pntlema  25285  pntlemb  25286  pntlemq  25290  pntlemr  25291  pntlemj  25292  pntlemf  25294  pntlemo  25296  pntleml  25300