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Theorem pntibndlem1 20754
Description: Lemma for pntibnd 20758. (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
StepHypRef Expression
1 pntibndlem1.l . . . 4  |-  L  =  ( ( 1  / 
4 )  /  ( A  +  3 ) )
2 4nn 9895 . . . . . 6  |-  4  e.  NN
3 nnrp 10379 . . . . . 6  |-  ( 4  e.  NN  ->  4  e.  RR+ )
4 rpreccl 10393 . . . . . 6  |-  ( 4  e.  RR+  ->  ( 1  /  4 )  e.  RR+ )
52, 3, 4mp2b 9 . . . . 5  |-  ( 1  /  4 )  e.  RR+
6 pntibndlem1.1 . . . . . 6  |-  ( ph  ->  A  e.  RR+ )
7 3nn 9894 . . . . . . 7  |-  3  e.  NN
8 nnrp 10379 . . . . . . 7  |-  ( 3  e.  NN  ->  3  e.  RR+ )
97, 8ax-mp 8 . . . . . 6  |-  3  e.  RR+
10 rpaddcl 10390 . . . . . 6  |-  ( ( A  e.  RR+  /\  3  e.  RR+ )  ->  ( A  +  3 )  e.  RR+ )
116, 9, 10sylancl 643 . . . . 5  |-  ( ph  ->  ( A  +  3 )  e.  RR+ )
12 rpdivcl 10392 . . . . 5  |-  ( ( ( 1  /  4
)  e.  RR+  /\  ( A  +  3 )  e.  RR+ )  ->  (
( 1  /  4
)  /  ( A  +  3 ) )  e.  RR+ )
135, 11, 12sylancr 644 . . . 4  |-  ( ph  ->  ( ( 1  / 
4 )  /  ( A  +  3 ) )  e.  RR+ )
141, 13syl5eqel 2380 . . 3  |-  ( ph  ->  L  e.  RR+ )
1514rpred 10406 . 2  |-  ( ph  ->  L  e.  RR )
1614rpgt0d 10409 . 2  |-  ( ph  ->  0  <  L )
17 rpcn 10378 . . . . . . 7  |-  ( ( 1  /  4 )  e.  RR+  ->  ( 1  /  4 )  e.  CC )
185, 17ax-mp 8 . . . . . 6  |-  ( 1  /  4 )  e.  CC
1918div1i 9504 . . . . 5  |-  ( ( 1  /  4 )  /  1 )  =  ( 1  /  4
)
20 rpre 10376 . . . . . . 7  |-  ( ( 1  /  4 )  e.  RR+  ->  ( 1  /  4 )  e.  RR )
215, 20mp1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
22 3re 9833 . . . . . . 7  |-  3  e.  RR
2322a1i 10 . . . . . 6  |-  ( ph  ->  3  e.  RR )
2411rpred 10406 . . . . . 6  |-  ( ph  ->  ( A  +  3 )  e.  RR )
25 1lt4 9907 . . . . . . . . 9  |-  1  <  4
26 4re 9835 . . . . . . . . . 10  |-  4  e.  RR
27 4pos 9848 . . . . . . . . . 10  |-  0  <  4
28 recgt1 9668 . . . . . . . . . 10  |-  ( ( 4  e.  RR  /\  0  <  4 )  -> 
( 1  <  4  <->  ( 1  /  4 )  <  1 ) )
2926, 27, 28mp2an 653 . . . . . . . . 9  |-  ( 1  <  4  <->  ( 1  /  4 )  <  1 )
3025, 29mpbi 199 . . . . . . . 8  |-  ( 1  /  4 )  <  1
31 1lt3 9904 . . . . . . . 8  |-  1  <  3
325, 20ax-mp 8 . . . . . . . . 9  |-  ( 1  /  4 )  e.  RR
33 1re 8853 . . . . . . . . 9  |-  1  e.  RR
3432, 33, 22lttri 8961 . . . . . . . 8  |-  ( ( ( 1  /  4
)  <  1  /\  1  <  3 )  -> 
( 1  /  4
)  <  3 )
3530, 31, 34mp2an 653 . . . . . . 7  |-  ( 1  /  4 )  <  3
3635a1i 10 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  <  3 )
37 ltaddrp 10402 . . . . . . . 8  |-  ( ( 3  e.  RR  /\  A  e.  RR+ )  -> 
3  <  ( 3  +  A ) )
3822, 6, 37sylancr 644 . . . . . . 7  |-  ( ph  ->  3  <  ( 3  +  A ) )
39 3cn 9834 . . . . . . . 8  |-  3  e.  CC
406rpcnd 10408 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
41 addcom 9014 . . . . . . . 8  |-  ( ( 3  e.  CC  /\  A  e.  CC )  ->  ( 3  +  A
)  =  ( A  +  3 ) )
4239, 40, 41sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 3  +  A
)  =  ( A  +  3 ) )
4338, 42breqtrd 4063 . . . . . 6  |-  ( ph  ->  3  <  ( A  +  3 ) )
4421, 23, 24, 36, 43lttrd 8993 . . . . 5  |-  ( ph  ->  ( 1  /  4
)  <  ( A  +  3 ) )
4519, 44syl5eqbr 4072 . . . 4  |-  ( ph  ->  ( ( 1  / 
4 )  /  1
)  <  ( A  +  3 ) )
4633a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR )
47 0lt1 9312 . . . . . 6  |-  0  <  1
4847a1i 10 . . . . 5  |-  ( ph  ->  0  <  1 )
4911rpregt0d 10412 . . . . 5  |-  ( ph  ->  ( ( A  + 
3 )  e.  RR  /\  0  <  ( A  +  3 ) ) )
50 ltdiv23 9663 . . . . 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 ) )
5121, 46, 48, 49, 50syl121anc 1187 . . . 4  |-  ( ph  ->  ( ( ( 1  /  4 )  / 
1 )  <  ( A  +  3 )  <-> 
( ( 1  / 
4 )  /  ( A  +  3 ) )  <  1 ) )
5245, 51mpbid 201 . . 3  |-  ( ph  ->  ( ( 1  / 
4 )  /  ( A  +  3 ) )  <  1 )
531, 52syl5eqbr 4072 . 2  |-  ( ph  ->  L  <  1 )
54 0xr 8894 . . 3  |-  0  e.  RR*
55 rexr 8893 . . . 4  |-  ( 1  e.  RR  ->  1  e.  RR* )
5633, 55ax-mp 8 . . 3  |-  1  e.  RR*
57 elioo2 10713 . . 3  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( L  e.  ( 0 (,) 1 )  <->  ( L  e.  RR  /\  0  < 
L  /\  L  <  1 ) ) )
5854, 56, 57mp2an 653 . 2  |-  ( L  e.  ( 0 (,) 1 )  <->  ( L  e.  RR  /\  0  < 
L  /\  L  <  1 ) )
5915, 16, 53, 58syl3anbrc 1136 1  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   3c3 9812   4c4 9813   RR+crp 10370   (,)cioo 10672  ψcchp 20346
This theorem is referenced by:  pntibndlem2a  20755  pntibndlem2  20756  pntibnd  20758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-rp 10371  df-ioo 10676
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