Theorem pntibndlem1 22956

Description: Lemma for pntibnd 22960. (Contributed by Mario Carneiro, 10-Apr-2016.)

Hypotheses

Ref	Expression
pntibnd.r	|- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
pntibndlem1.1	|- ( ph -> A e. RR+ )
pntibndlem1.l	|- L = ( ( 1 / 4 ) / ( A + 3 ) )

Assertion

Ref	Expression
pntibndlem1	|- ( ph -> L e. ( 0 (,) 1 ) )

Proof of Theorem pntibndlem1

Step	Hyp	Ref	Expression
1		pntibndlem1.l	. . . 4 |- L = ( ( 1 / 4 ) / ( A + 3 ) )
2		4nn 10584	. . . . . 6 |- 4 e. NN
3		nnrp 11103	. . . . . 6 |- ( 4 e. NN -> 4 e. RR+ )
4		rpreccl 11117	. . . . . 6 |- ( 4 e. RR+ -> ( 1 / 4 ) e. RR+ )
5	2, 3, 4	mp2b 10	. . . . 5 |- ( 1 / 4 ) e. RR+
6		pntibndlem1.1	. . . . . 6 |- ( ph -> A e. RR+ )
7		3nn 10583	. . . . . . 7 |- 3 e. NN
8		nnrp 11103	. . . . . . 7 |- ( 3 e. NN -> 3 e. RR+ )
9	7, 8	ax-mp 5	. . . . . 6 |- 3 e. RR+
10		rpaddcl 11114	. . . . . 6 |- ( ( A e. RR+ /\ 3 e. RR+ ) -> ( A + 3 ) e. RR+ )
11	6, 9, 10	sylancl 662	. . . . 5 |- ( ph -> ( A + 3 ) e. RR+ )
12		rpdivcl 11116	. . . . 5 |- ( ( ( 1 / 4 ) e. RR+ /\ ( A + 3 ) e. RR+ ) -> ( ( 1 / 4 ) / ( A + 3 ) ) e. RR+ )
13	5, 11, 12	sylancr 663	. . . 4 |- ( ph -> ( ( 1 / 4 ) / ( A + 3 ) ) e. RR+ )
14	1, 13	syl5eqel 2543	. . 3 |- ( ph -> L e. RR+ )
15	14	rpred 11130	. 2 |- ( ph -> L e. RR )
16	14	rpgt0d 11133	. 2 |- ( ph -> 0 < L )
17		rpcn 11102	. . . . . . 7 |- ( ( 1 / 4 ) e. RR+ -> ( 1 / 4 ) e. CC )
18	5, 17	ax-mp 5	. . . . . 6 |- ( 1 / 4 ) e. CC
19	18	div1i 10162	. . . . 5 |- ( ( 1 / 4 ) / 1 ) = ( 1 / 4 )
20		rpre 11100	. . . . . . 7 |- ( ( 1 / 4 ) e. RR+ -> ( 1 / 4 ) e. RR )
21	5, 20	mp1i 12	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR )
22		3re 10498	. . . . . . 7 |- 3 e. RR
23	22	a1i 11	. . . . . 6 |- ( ph -> 3 e. RR )
24	11	rpred 11130	. . . . . 6 |- ( ph -> ( A + 3 ) e. RR )
25		1lt4 10596	. . . . . . . . 9 |- 1 < 4
26		4re 10501	. . . . . . . . . 10 |- 4 e. RR
27		4pos 10520	. . . . . . . . . 10 |- 0 < 4
28		recgt1 10331	. . . . . . . . . 10 |- ( ( 4 e. RR /\ 0 < 4 ) -> ( 1 < 4 <-> ( 1 / 4 ) < 1 ) )
29	26, 27, 28	mp2an 672	. . . . . . . . 9 |- ( 1 < 4 <-> ( 1 / 4 ) < 1 )
30	25, 29	mpbi 208	. . . . . . . 8 |- ( 1 / 4 ) < 1
31		1lt3 10593	. . . . . . . 8 |- 1 < 3
32	5, 20	ax-mp 5	. . . . . . . . 9 |- ( 1 / 4 ) e. RR
33		1re 9488	. . . . . . . . 9 |- 1 e. RR
34	32, 33, 22	lttri 9603	. . . . . . . 8 |- ( ( ( 1 / 4 ) < 1 /\ 1 < 3 ) -> ( 1 / 4 ) < 3 )
35	30, 31, 34	mp2an 672	. . . . . . 7 |- ( 1 / 4 ) < 3
36	35	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < 3 )
37		ltaddrp 11126	. . . . . . . 8 |- ( ( 3 e. RR /\ A e. RR+ ) -> 3 < ( 3 + A ) )
38	22, 6, 37	sylancr 663	. . . . . . 7 |- ( ph -> 3 < ( 3 + A ) )
39		3cn 10499	. . . . . . . 8 |- 3 e. CC
40	6	rpcnd 11132	. . . . . . . 8 |- ( ph -> A e. CC )
41		addcom 9658	. . . . . . . 8 |- ( ( 3 e. CC /\ A e. CC ) -> ( 3 + A ) = ( A + 3 ) )
42	39, 40, 41	sylancr 663	. . . . . . 7 |- ( ph -> ( 3 + A ) = ( A + 3 ) )
43	38, 42	breqtrd 4416	. . . . . 6 |- ( ph -> 3 < ( A + 3 ) )
44	21, 23, 24, 36, 43	lttrd 9635	. . . . 5 |- ( ph -> ( 1 / 4 ) < ( A + 3 ) )
45	19, 44	syl5eqbr 4425	. . . 4 |- ( ph -> ( ( 1 / 4 ) / 1 ) < ( A + 3 ) )
46	33	a1i 11	. . . . 5 |- ( ph -> 1 e. RR )
47		0lt1 9965	. . . . . 6 |- 0 < 1
48	47	a1i 11	. . . . 5 |- ( ph -> 0 < 1 )
49	11	rpregt0d 11136	. . . . 5 |- ( ph -> ( ( A + 3 ) e. RR /\ 0 < ( A + 3 ) ) )
50		ltdiv23 10326	. . . . 5 |- ( ( ( 1 / 4 ) e. RR /\ ( 1 e. RR /\ 0 < 1 ) /\ ( ( A + 3 ) e. RR /\ 0 < ( A + 3 ) ) ) -> ( ( ( 1 / 4 ) / 1 ) < ( A + 3 ) <-> ( ( 1 / 4 ) / ( A + 3 ) ) < 1 ) )
51	21, 46, 48, 49, 50	syl121anc 1224	. . . 4 |- ( ph -> ( ( ( 1 / 4 ) / 1 ) < ( A + 3 ) <-> ( ( 1 / 4 ) / ( A + 3 ) ) < 1 ) )
52	45, 51	mpbid 210	. . 3 |- ( ph -> ( ( 1 / 4 ) / ( A + 3 ) ) < 1 )
53	1, 52	syl5eqbr 4425	. 2 |- ( ph -> L < 1 )
54		0xr 9533	. . 3 |- 0 e. RR*
55	33	rexri 9539	. . 3 |- 1 e. RR*
56		elioo2 11444	. . 3 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( L e. ( 0 (,) 1 ) <-> ( L e. RR /\ 0 < L /\ L < 1 ) ) )
57	54, 55, 56	mp2an 672	. 2 |- ( L e. ( 0 (,) 1 ) <-> ( L e. RR /\ 0 < L /\ L < 1 ) )
58	15, 16, 53, 57	syl3anbrc 1172	1 |- ( ph -> L e. ( 0 (,) 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   class class class wbr 4392   |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385   1c1 9386   + caddc 9388   RR*cxr 9520   < clt 9521   - cmin 9698   / cdiv 10096   NNcn 10425   3c3 10475   4c4 10476   RR+crp 11094   (,)cioo 11403  ψcchp 22548

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-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-rp 11095  df-ioo 11407

This theorem is referenced by:  pntibndlem2a  22957  pntibndlem2  22958  pntibnd  22960