Theorem pntibndlem1 23972

Description: Lemma for pntibnd 23976. (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 10691	. . . . . 6 |- 4 e. NN
3		nnrp 11230	. . . . . 6 |- ( 4 e. NN -> 4 e. RR+ )
4		rpreccl 11245	. . . . . 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 10690	. . . . . . 7 |- 3 e. NN
8		nnrp 11230	. . . . . . 7 |- ( 3 e. NN -> 3 e. RR+ )
9	7, 8	ax-mp 5	. . . . . 6 |- 3 e. RR+
10		rpaddcl 11242	. . . . . 6 |- ( ( A e. RR+ /\ 3 e. RR+ ) -> ( A + 3 ) e. RR+ )
11	6, 9, 10	sylancl 660	. . . . 5 |- ( ph -> ( A + 3 ) e. RR+ )
12		rpdivcl 11244	. . . . 5 |- ( ( ( 1 / 4 ) e. RR+ /\ ( A + 3 ) e. RR+ ) -> ( ( 1 / 4 ) / ( A + 3 ) ) e. RR+ )
13	5, 11, 12	sylancr 661	. . . 4 |- ( ph -> ( ( 1 / 4 ) / ( A + 3 ) ) e. RR+ )
14	1, 13	syl5eqel 2546	. . 3 |- ( ph -> L e. RR+ )
15	14	rpred 11259	. 2 |- ( ph -> L e. RR )
16	14	rpgt0d 11262	. 2 |- ( ph -> 0 < L )
17		rpcn 11229	. . . . . . 7 |- ( ( 1 / 4 ) e. RR+ -> ( 1 / 4 ) e. CC )
18	5, 17	ax-mp 5	. . . . . 6 |- ( 1 / 4 ) e. CC
19	18	div1i 10268	. . . . 5 |- ( ( 1 / 4 ) / 1 ) = ( 1 / 4 )
20		rpre 11227	. . . . . . 7 |- ( ( 1 / 4 ) e. RR+ -> ( 1 / 4 ) e. RR )
21	5, 20	mp1i 12	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR )
22		3re 10605	. . . . . . 7 |- 3 e. RR
23	22	a1i 11	. . . . . 6 |- ( ph -> 3 e. RR )
24	11	rpred 11259	. . . . . 6 |- ( ph -> ( A + 3 ) e. RR )
25		1lt4 10703	. . . . . . . . 9 |- 1 < 4
26		4re 10608	. . . . . . . . . 10 |- 4 e. RR
27		4pos 10627	. . . . . . . . . 10 |- 0 < 4
28		recgt1 10436	. . . . . . . . . 10 |- ( ( 4 e. RR /\ 0 < 4 ) -> ( 1 < 4 <-> ( 1 / 4 ) < 1 ) )
29	26, 27, 28	mp2an 670	. . . . . . . . 9 |- ( 1 < 4 <-> ( 1 / 4 ) < 1 )
30	25, 29	mpbi 208	. . . . . . . 8 |- ( 1 / 4 ) < 1
31		1lt3 10700	. . . . . . . 8 |- 1 < 3
32	5, 20	ax-mp 5	. . . . . . . . 9 |- ( 1 / 4 ) e. RR
33		1re 9584	. . . . . . . . 9 |- 1 e. RR
34	32, 33, 22	lttri 9699	. . . . . . . 8 |- ( ( ( 1 / 4 ) < 1 /\ 1 < 3 ) -> ( 1 / 4 ) < 3 )
35	30, 31, 34	mp2an 670	. . . . . . 7 |- ( 1 / 4 ) < 3
36	35	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < 3 )
37		ltaddrp 11254	. . . . . . . 8 |- ( ( 3 e. RR /\ A e. RR+ ) -> 3 < ( 3 + A ) )
38	22, 6, 37	sylancr 661	. . . . . . 7 |- ( ph -> 3 < ( 3 + A ) )
39		3cn 10606	. . . . . . . 8 |- 3 e. CC
40	6	rpcnd 11261	. . . . . . . 8 |- ( ph -> A e. CC )
41		addcom 9755	. . . . . . . 8 |- ( ( 3 e. CC /\ A e. CC ) -> ( 3 + A ) = ( A + 3 ) )
42	39, 40, 41	sylancr 661	. . . . . . 7 |- ( ph -> ( 3 + A ) = ( A + 3 ) )
43	38, 42	breqtrd 4463	. . . . . 6 |- ( ph -> 3 < ( A + 3 ) )
44	21, 23, 24, 36, 43	lttrd 9732	. . . . 5 |- ( ph -> ( 1 / 4 ) < ( A + 3 ) )
45	19, 44	syl5eqbr 4472	. . . 4 |- ( ph -> ( ( 1 / 4 ) / 1 ) < ( A + 3 ) )
46	33	a1i 11	. . . . 5 |- ( ph -> 1 e. RR )
47		0lt1 10071	. . . . . 6 |- 0 < 1
48	47	a1i 11	. . . . 5 |- ( ph -> 0 < 1 )
49	11	rpregt0d 11265	. . . . 5 |- ( ph -> ( ( A + 3 ) e. RR /\ 0 < ( A + 3 ) ) )
50		ltdiv23 10431	. . . . 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 1231	. . . 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 4472	. 2 |- ( ph -> L < 1 )
54		0xr 9629	. . 3 |- 0 e. RR*
55	33	rexri 9635	. . 3 |- 1 e. RR*
56		elioo2 11573	. . 3 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( L e. ( 0 (,) 1 ) <-> ( L e. RR /\ 0 < L /\ L < 1 ) ) )
57	54, 55, 56	mp2an 670	. 2 |- ( L e. ( 0 (,) 1 ) <-> ( L e. RR /\ 0 < L /\ L < 1 ) )
58	15, 16, 53, 57	syl3anbrc 1178	1 |- ( ph -> L e. ( 0 (,) 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   class class class wbr 4439   |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   RR*cxr 9616   < clt 9617   - cmin 9796   / cdiv 10202   NNcn 10531   3c3 10582   4c4 10583   RR+crp 11221   (,)cioo 11532  ψcchp 23564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-rp 11222  df-ioo 11536

This theorem is referenced by:  pntibndlem2a  23973  pntibndlem2  23974  pntibnd  23976