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Theorem leibpilem1 22994
Description: Lemma for leibpi 22996. (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 10788 . . . . . . 7  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
21biimpi 194 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  e.  NN  \/  N  =  0 ) )
32ord 377 . . . . 5  |-  ( N  e.  NN0  ->  ( -.  N  e.  NN  ->  N  =  0 ) )
43con1d 124 . . . 4  |-  ( N  e.  NN0  ->  ( -.  N  =  0  ->  N  e.  NN )
)
54imp 429 . . 3  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  N  e.  NN )
65adantrr 716 . 2  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  ->  N  e.  NN )
7 nn0z 10878 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
87adantr 465 . . . . . 6  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  N  e.  ZZ )
9 odd2np1 13896 . . . . . 6  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
108, 9syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
11 zcn 10860 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  CC )
12 2cn 10597 . . . . . . . . . . . . 13  |-  2  e.  CC
13 mulcl 9567 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  n  e.  CC )  ->  ( 2  x.  n
)  e.  CC )
1412, 13mpan 670 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
2  x.  n )  e.  CC )
15 ax-1cn 9541 . . . . . . . . . . . 12  |-  1  e.  CC
16 pncan 9817 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
1714, 15, 16sylancl 662 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  (
( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
1817oveq1d 6292 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  / 
2 ) )
19 2ne0 10619 . . . . . . . . . . 11  |-  2  =/=  0
20 divcan3 10222 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  n
)  /  2 )  =  n )
2112, 19, 20mp3an23 1311 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  /  2 )  =  n )
2218, 21eqtrd 2503 . . . . . . . . 9  |-  ( n  e.  CC  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
2311, 22syl 16 . . . . . . . 8  |-  ( n  e.  ZZ  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
24 id 22 . . . . . . . 8  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
2523, 24eqeltrd 2550 . . . . . . 7  |-  ( n  e.  ZZ  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  e.  ZZ )
26 oveq1 6284 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  -  1 )  =  ( N  - 
1 ) )
2726oveq1d 6292 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( N  -  1 )  / 
2 ) )
2827eleq1d 2531 . . . . . . 7  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
)  e.  ZZ  <->  ( ( N  -  1 )  /  2 )  e.  ZZ ) )
2925, 28syl5ibcom 220 . . . . . 6  |-  ( n  e.  ZZ  ->  (
( ( 2  x.  n )  +  1 )  =  N  -> 
( ( N  - 
1 )  /  2
)  e.  ZZ ) )
3029rexlimiv 2944 . . . . 5  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  ->  (
( N  -  1 )  /  2 )  e.  ZZ )
3110, 30syl6bi 228 . . . 4  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  ( -.  2  ||  N  ->  (
( N  -  1 )  /  2 )  e.  ZZ ) )
3231impr 619 . . 3  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( ( N  - 
1 )  /  2
)  e.  ZZ )
33 nnm1nn0 10828 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
346, 33syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  NN0 )
3534nn0red 10844 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  RR )
3634nn0ge0d 10846 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( N  -  1 ) )
37 2re 10596 . . . . 5  |-  2  e.  RR
3837a1i 11 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
2  e.  RR )
39 2pos 10618 . . . . 5  |-  0  <  2
4039a1i 11 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <  2 )
41 divge0 10402 . . . 4  |-  ( ( ( ( N  - 
1 )  e.  RR  /\  0  <_  ( N  -  1 ) )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
4235, 36, 38, 40, 41syl22anc 1224 . . 3  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
43 elnn0z 10868 . . 3  |-  ( ( ( N  -  1 )  /  2 )  e.  NN0  <->  ( ( ( N  -  1 )  /  2 )  e.  ZZ  /\  0  <_ 
( ( N  - 
1 )  /  2
) ) )
4432, 42, 43sylanbrc 664 . 2  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( ( N  - 
1 )  /  2
)  e.  NN0 )
456, 44jca 532 1  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  e.  NN  /\  ( ( N  - 
1 )  /  2
)  e.  NN0 )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   class class class wbr 4442  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    < clt 9619    <_ cle 9620    - cmin 9796    / cdiv 10197   NNcn 10527   2c2 10576   NN0cn0 10786   ZZcz 10855    || cdivides 13838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-dvds 13839
This theorem is referenced by:  leibpilem2  22995  leibpi  22996
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