Theorem pntlemc 22728

Description: Lemma for pnt 22747. 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 22727	. . . . 5 |- ( ph -> ( L e. RR+ /\ D e. RR+ /\ F e. RR+ ) )
10	9	simp2d 994	. . . 4 |- ( ph -> D e. RR+ )
11	2, 10	rpdivcld 11031	. . 3 |- ( ph -> ( U / D ) e. RR+ )
12	1, 11	syl5eqel 2517	. 2 |- ( ph -> E e. RR+ )
13		pntlem1.k	. . 3 |- K = ( exp ` ( B / E ) )
14	5, 12	rpdivcld 11031	. . . . 5 |- ( ph -> ( B / E ) e. RR+ )
15	14	rpred 11014	. . . 4 |- ( ph -> ( B / E ) e. RR )
16	15	rpefcld 13371	. . 3 |- ( ph -> ( exp ` ( B / E ) ) e. RR+ )
17	13, 16	syl5eqel 2517	. 2 |- ( ph -> K e. RR+ )
18	12	rpred 11014	. . . 4 |- ( ph -> E e. RR )
19	12	rpgt0d 11017	. . . 4 |- ( ph -> 0 < E )
20	2	rpred 11014	. . . . . . . 8 |- ( ph -> U e. RR )
21	4	rpred 11014	. . . . . . . 8 |- ( ph -> A e. RR )
22	10	rpred 11014	. . . . . . . 8 |- ( ph -> D e. RR )
23		pntlem1.u2	. . . . . . . 8 |- ( ph -> U <_ A )
24	21	ltp1d 10250	. . . . . . . . 9 |- ( ph -> A < ( A + 1 ) )
25	24, 7	syl6breqr 4320	. . . . . . . 8 |- ( ph -> A < D )
26	20, 21, 22, 23, 25	lelttrd 9516	. . . . . . 7 |- ( ph -> U < D )
27	10	rpcnd 11016	. . . . . . . 8 |- ( ph -> D e. CC )
28	27	mulid1d 9390	. . . . . . 7 |- ( ph -> ( D x. 1 ) = D )
29	26, 28	breqtrrd 4306	. . . . . 6 |- ( ph -> U < ( D x. 1 ) )
30		1red 9388	. . . . . . 7 |- ( ph -> 1 e. RR )
31	20, 30, 10	ltdivmuld 11061	. . . . . 6 |- ( ph -> ( ( U / D ) < 1 <-> U < ( D x. 1 ) ) )
32	29, 31	mpbird 232	. . . . 5 |- ( ph -> ( U / D ) < 1 )
33	1, 32	syl5eqbr 4313	. . . 4 |- ( ph -> E < 1 )
34		0xr 9417	. . . . 5 |- 0 e. RR*
35		1re 9372	. . . . . 6 |- 1 e. RR
36	35	rexri 9423	. . . . 5 |- 1 e. RR*
37		elioo2 11328	. . . . 5 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) ) )
38	34, 36, 37	mp2an 665	. . . 4 |- ( E e. ( 0 (,) 1 ) <-> ( E e. RR /\ 0 < E /\ E < 1 ) )
39	18, 19, 33, 38	syl3anbrc 1165	. . 3 |- ( ph -> E e. ( 0 (,) 1 ) )
40		efgt1 13382	. . . . 5 |- ( ( B / E ) e. RR+ -> 1 < ( exp ` ( B / E ) ) )
41	14, 40	syl 16	. . . 4 |- ( ph -> 1 < ( exp ` ( B / E ) ) )
42	41, 13	syl6breqr 4320	. . 3 |- ( ph -> 1 < K )
43		ltaddrp 11010	. . . . . . . 8 |- ( ( 1 e. RR /\ A e. RR+ ) -> 1 < ( 1 + A ) )
44	35, 4, 43	sylancr 656	. . . . . . 7 |- ( ph -> 1 < ( 1 + A ) )
45	2	rpcnne0d 11023	. . . . . . . 8 |- ( ph -> ( U e. CC /\ U =/= 0 ) )
46		divid 10008	. . . . . . . 8 |- ( ( U e. CC /\ U =/= 0 ) -> ( U / U ) = 1 )
47	45, 46	syl 16	. . . . . . 7 |- ( ph -> ( U / U ) = 1 )
48	4	rpcnd 11016	. . . . . . . . 9 |- ( ph -> A e. CC )
49		ax-1cn 9327	. . . . . . . . 9 |- 1 e. CC
50		addcom 9542	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
51	48, 49, 50	sylancl 655	. . . . . . . 8 |- ( ph -> ( A + 1 ) = ( 1 + A ) )
52	7, 51	syl5eq 2477	. . . . . . 7 |- ( ph -> D = ( 1 + A ) )
53	44, 47, 52	3brtr4d 4310	. . . . . 6 |- ( ph -> ( U / U ) < D )
54	20, 2, 10, 53	ltdiv23d 11070	. . . . 5 |- ( ph -> ( U / D ) < U )
55	1, 54	syl5eqbr 4313	. . . 4 |- ( ph -> E < U )
56		difrp 11011	. . . . 5 |- ( ( E e. RR /\ U e. RR ) -> ( E < U <-> ( U - E ) e. RR+ ) )
57	18, 20, 56	syl2anc 654	. . . 4 |- ( ph -> ( E < U <-> ( U - E ) e. RR+ ) )
58	55, 57	mpbid 210	. . 3 |- ( ph -> ( U - E ) e. RR+ )
59	39, 42, 58	3jca 1161	. 2 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
60	12, 17, 59	3jca 1161	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 958   = wceq 1362   e. wcel 1755   =/= wne 2596   class class class wbr 4280   e. cmpt 4338   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270   + caddc 9272   x. cmul 9274   RR*cxr 9404   < clt 9405   <_ cle 9406   - cmin 9582   / cdiv 9980   2c2 10358   3c3 10359  ;cdc 10742   RR+crp 10978   (,)cioo 11287   ^cexp 11848   expce 13329  ψcchp 22314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-rp 10979  df-ioo 11291  df-ico 11293  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335

This theorem is referenced by:  pntlema  22729  pntlemb  22730  pntlemg  22731  pntlemh  22732  pntlemq  22734  pntlemr  22735  pntlemj  22736  pntlemi  22737  pntlemf  22738  pntlemo  22740  pntleme  22741  pntlemp  22743