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Theorem leibpilem1 20252
Description: Lemma for leibpi 20254. (Contributed by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
leibpilem1  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  e.  NN  /\  ( ( N  - 
1 )  /  2
)  e.  NN0 )
)

Proof of Theorem leibpilem1
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 elnn0 9983 . . . . . . 7  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
21biimpi 186 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  e.  NN  \/  N  =  0 ) )
32ord 366 . . . . 5  |-  ( N  e.  NN0  ->  ( -.  N  e.  NN  ->  N  =  0 ) )
43con1d 116 . . . 4  |-  ( N  e.  NN0  ->  ( -.  N  =  0  ->  N  e.  NN )
)
54imp 418 . . 3  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  N  e.  NN )
65adantrr 697 . 2  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  ->  N  e.  NN )
7 nn0z 10062 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
87adantr 451 . . . . . 6  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  N  e.  ZZ )
9 odd2np1 12603 . . . . . 6  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
108, 9syl 15 . . . . 5  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
11 zcn 10045 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  CC )
12 2cn 9832 . . . . . . . . . . . . 13  |-  2  e.  CC
13 mulcl 8837 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  n  e.  CC )  ->  ( 2  x.  n
)  e.  CC )
1412, 13mpan 651 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
2  x.  n )  e.  CC )
15 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
16 pncan 9073 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
1714, 15, 16sylancl 643 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  (
( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
1817oveq1d 5889 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  / 
2 ) )
19 2ne0 9845 . . . . . . . . . . 11  |-  2  =/=  0
20 divcan3 9464 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  n
)  /  2 )  =  n )
2112, 19, 20mp3an23 1269 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  /  2 )  =  n )
2218, 21eqtrd 2328 . . . . . . . . 9  |-  ( n  e.  CC  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
2311, 22syl 15 . . . . . . . 8  |-  ( n  e.  ZZ  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
24 id 19 . . . . . . . 8  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
2523, 24eqeltrd 2370 . . . . . . 7  |-  ( n  e.  ZZ  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  e.  ZZ )
26 oveq1 5881 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  -  1 )  =  ( N  - 
1 ) )
2726oveq1d 5889 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( N  -  1 )  / 
2 ) )
2827eleq1d 2362 . . . . . . 7  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
)  e.  ZZ  <->  ( ( N  -  1 )  /  2 )  e.  ZZ ) )
2925, 28syl5ibcom 211 . . . . . 6  |-  ( n  e.  ZZ  ->  (
( ( 2  x.  n )  +  1 )  =  N  -> 
( ( N  - 
1 )  /  2
)  e.  ZZ ) )
3029rexlimiv 2674 . . . . 5  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  ->  (
( N  -  1 )  /  2 )  e.  ZZ )
3110, 30syl6bi 219 . . . 4  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  ( -.  2  ||  N  ->  (
( N  -  1 )  /  2 )  e.  ZZ ) )
3231impr 602 . . 3  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( ( N  - 
1 )  /  2
)  e.  ZZ )
33 nnm1nn0 10021 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
346, 33syl 15 . . . . 5  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  NN0 )
3534nn0red 10035 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  RR )
3634nn0ge0d 10037 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( N  -  1 ) )
37 2re 9831 . . . . 5  |-  2  e.  RR
3837a1i 10 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
2  e.  RR )
39 2pos 9844 . . . . 5  |-  0  <  2
4039a1i 10 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <  2 )
41 divge0 9641 . . . 4  |-  ( ( ( ( N  - 
1 )  e.  RR  /\  0  <_  ( N  -  1 ) )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
4235, 36, 38, 40, 41syl22anc 1183 . . 3  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
43 elnn0z 10052 . . 3  |-  ( ( ( N  -  1 )  /  2 )  e.  NN0  <->  ( ( ( N  -  1 )  /  2 )  e.  ZZ  /\  0  <_ 
( ( N  - 
1 )  /  2
) ) )
4432, 42, 43sylanbrc 645 . 2  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( ( N  - 
1 )  /  2
)  e.  NN0 )
456, 44jca 518 1  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  e.  NN  /\  ( ( N  - 
1 )  /  2
)  e.  NN0 )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  leibpilem2  20253  leibpi  20254
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-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-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-n0 9982  df-z 10041  df-dvds 12548
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