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Theorem leibpilem1 22334
Description: Lemma for leibpi 22336. (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 10580 . . . . . . 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 10668 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
87adantr 465 . . . . . 6  |-  ( ( N  e.  NN0  /\  -.  N  =  0
)  ->  N  e.  ZZ )
9 odd2np1 13591 . . . . . 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 10650 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  CC )
12 2cn 10391 . . . . . . . . . . . . 13  |-  2  e.  CC
13 mulcl 9365 . . . . . . . . . . . . 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 9339 . . . . . . . . . . . 12  |-  1  e.  CC
16 pncan 9615 . . . . . . . . . . . 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 6105 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  / 
2 ) )
19 2ne0 10413 . . . . . . . . . . 11  |-  2  =/=  0
20 divcan3 10017 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  n
)  /  2 )  =  n )
2112, 19, 20mp3an23 1306 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  /  2 )  =  n )
2218, 21eqtrd 2474 . . . . . . . . 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 2516 . . . . . . 7  |-  ( n  e.  ZZ  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  e.  ZZ )
26 oveq1 6097 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  -  1 )  =  ( N  - 
1 ) )
2726oveq1d 6105 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( N  -  1 )  / 
2 ) )
2827eleq1d 2508 . . . . . . 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 2834 . . . . 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 10620 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
346, 33syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  NN0 )
3534nn0red 10636 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
( N  -  1 )  e.  RR )
3634nn0ge0d 10638 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( N  -  1 ) )
37 2re 10390 . . . . 5  |-  2  e.  RR
3837a1i 11 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
2  e.  RR )
39 2pos 10412 . . . . 5  |-  0  <  2
4039a1i 11 . . . 4  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <  2 )
41 divge0 10197 . . . 4  |-  ( ( ( ( N  - 
1 )  e.  RR  /\  0  <_  ( N  -  1 ) )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
4235, 36, 38, 40, 41syl22anc 1219 . . 3  |-  ( ( N  e.  NN0  /\  ( -.  N  = 
0  /\  -.  2  ||  N ) )  -> 
0  <_  ( ( N  -  1 )  /  2 ) )
43 elnn0z 10658 . . 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 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715   class class class wbr 4291  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594    / cdiv 9992   NNcn 10321   2c2 10370   NN0cn0 10578   ZZcz 10645    || cdivides 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-dvds 13535
This theorem is referenced by:  leibpilem2  22335  leibpi  22336
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