MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsdir2lem2 Structured version   Unicode version

Theorem lgsdir2lem2 23725
Description: Lemma for lgsdir2 23729. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypotheses
Ref Expression
lgsdir2lem2.1  |-  ( K  e.  ZZ  /\  2  ||  ( K  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... K
)  ->  ( A  mod  8 )  e.  S
) ) )
lgsdir2lem2.2  |-  M  =  ( K  +  1 )
lgsdir2lem2.3  |-  N  =  ( M  +  1 )
lgsdir2lem2.4  |-  N  e.  S
Assertion
Ref Expression
lgsdir2lem2  |-  ( N  e.  ZZ  /\  2  ||  ( N  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) ) )

Proof of Theorem lgsdir2lem2
StepHypRef Expression
1 lgsdir2lem2.3 . . 3  |-  N  =  ( M  +  1 )
2 lgsdir2lem2.2 . . . . 5  |-  M  =  ( K  +  1 )
3 lgsdir2lem2.1 . . . . . . 7  |-  ( K  e.  ZZ  /\  2  ||  ( K  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... K
)  ->  ( A  mod  8 )  e.  S
) ) )
43simp1i 1005 . . . . . 6  |-  K  e.  ZZ
5 peano2z 10926 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
64, 5ax-mp 5 . . . . 5  |-  ( K  +  1 )  e.  ZZ
72, 6eqeltri 2541 . . . 4  |-  M  e.  ZZ
8 peano2z 10926 . . . 4  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
97, 8ax-mp 5 . . 3  |-  ( M  +  1 )  e.  ZZ
101, 9eqeltri 2541 . 2  |-  N  e.  ZZ
113simp2i 1006 . . . 4  |-  2  ||  ( K  +  1 )
12 2z 10917 . . . . 5  |-  2  e.  ZZ
13 dvdsadd 14036 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( 2  ||  ( K  +  1
)  <->  2  ||  (
2  +  ( K  +  1 ) ) ) )
1412, 6, 13mp2an 672 . . . 4  |-  ( 2 
||  ( K  + 
1 )  <->  2  ||  ( 2  +  ( K  +  1 ) ) )
1511, 14mpbi 208 . . 3  |-  2  ||  ( 2  +  ( K  +  1 ) )
16 zcn 10890 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
174, 16ax-mp 5 . . . . . . . . . 10  |-  K  e.  CC
18 ax-1cn 9567 . . . . . . . . . 10  |-  1  e.  CC
1917, 18addcomi 9788 . . . . . . . . 9  |-  ( K  +  1 )  =  ( 1  +  K
)
202, 19eqtri 2486 . . . . . . . 8  |-  M  =  ( 1  +  K
)
2120oveq1i 6306 . . . . . . 7  |-  ( M  +  1 )  =  ( ( 1  +  K )  +  1 )
221, 21eqtri 2486 . . . . . 6  |-  N  =  ( ( 1  +  K )  +  1 )
23 df-2 10615 . . . . . . . 8  |-  2  =  ( 1  +  1 )
2423oveq1i 6306 . . . . . . 7  |-  ( 2  +  K )  =  ( ( 1  +  1 )  +  K
)
2518, 17, 18add32i 9816 . . . . . . 7  |-  ( ( 1  +  K )  +  1 )  =  ( ( 1  +  1 )  +  K
)
2624, 25eqtr4i 2489 . . . . . 6  |-  ( 2  +  K )  =  ( ( 1  +  K )  +  1 )
2722, 26eqtr4i 2489 . . . . 5  |-  N  =  ( 2  +  K
)
2827oveq1i 6306 . . . 4  |-  ( N  +  1 )  =  ( ( 2  +  K )  +  1 )
29 2cn 10627 . . . . 5  |-  2  e.  CC
3029, 17, 18addassi 9621 . . . 4  |-  ( ( 2  +  K )  +  1 )  =  ( 2  +  ( K  +  1 ) )
3128, 30eqtri 2486 . . 3  |-  ( N  +  1 )  =  ( 2  +  ( K  +  1 ) )
3215, 31breqtrri 4481 . 2  |-  2  ||  ( N  +  1 )
33 elfzuz2 11716 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
34 fzm1 11784 . . . . 5  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  <->  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N ) ) )
3533, 34syl 16 . . . 4  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  (
( A  mod  8
)  e.  ( 0 ... N )  <->  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N ) ) )
3635ibi 241 . . 3  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  (
( A  mod  8
)  e.  ( 0 ... ( N  - 
1 ) )  \/  ( A  mod  8
)  =  N ) )
37 elfzuz2 11716 . . . . . . . 8  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  M  e.  ( ZZ>= `  0 )
)
38 fzm1 11784 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( ( A  mod  8 )  e.  ( 0 ... M
)  <->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M ) ) )
3937, 38syl 16 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  (
( A  mod  8
)  e.  ( 0 ... M )  <->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M ) ) )
4039ibi 241 . . . . . 6  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
41 zcn 10890 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
4210, 41ax-mp 5 . . . . . . . 8  |-  N  e.  CC
43 zcn 10890 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
447, 43ax-mp 5 . . . . . . . 8  |-  M  e.  CC
4518, 44addcomi 9788 . . . . . . . . 9  |-  ( 1  +  M )  =  ( M  +  1 )
4645, 1eqtr4i 2489 . . . . . . . 8  |-  ( 1  +  M )  =  N
4742, 18, 44, 46subaddrii 9928 . . . . . . 7  |-  ( N  -  1 )  =  M
4847oveq2i 6307 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... M
)
4940, 48eleq2s 2565 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
5020eqcomi 2470 . . . . . . . . . 10  |-  ( 1  +  K )  =  M
5144, 18, 17, 50subaddrii 9928 . . . . . . . . 9  |-  ( M  -  1 )  =  K
5251oveq2i 6307 . . . . . . . 8  |-  ( 0 ... ( M  - 
1 ) )  =  ( 0 ... K
)
5352eleq2i 2535 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  <->  ( A  mod  8 )  e.  ( 0 ... K ) )
543simp3i 1007 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... K
)  ->  ( A  mod  8 )  e.  S
) )
5553, 54syl5bi 217 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  ->  ( A  mod  8 )  e.  S
) )
56 2nn 10714 . . . . . . . . . . 11  |-  2  e.  NN
57 8nn 10720 . . . . . . . . . . 11  |-  8  e.  NN
58 4z 10919 . . . . . . . . . . . . . 14  |-  4  e.  ZZ
59 dvdsmul2 14018 . . . . . . . . . . . . . 14  |-  ( ( 4  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( 4  x.  2 ) )
6058, 12, 59mp2an 672 . . . . . . . . . . . . 13  |-  2  ||  ( 4  x.  2 )
61 4t2e8 10710 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6260, 61breqtri 4479 . . . . . . . . . . . 12  |-  2  ||  8
63 dvdsmod 14055 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  /\  2  ||  8 )  ->  ( 2  ||  ( A  mod  8
)  <->  2  ||  A
) )
6462, 63mpan2 671 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6556, 57, 64mp3an12 1314 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6665notbid 294 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( -.  2  ||  ( A  mod  8 )  <->  -.  2  ||  A ) )
6766biimpar 485 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  2  ||  ( A  mod  8
) )
6811, 2breqtrri 4481 . . . . . . . . 9  |-  2  ||  M
69 id 22 . . . . . . . . 9  |-  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  =  M )
7068, 69syl5breqr 4492 . . . . . . . 8  |-  ( ( A  mod  8 )  =  M  ->  2  ||  ( A  mod  8
) )
7167, 70nsyl 121 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  ( A  mod  8 )  =  M )
7271pm2.21d 106 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  e.  S
) )
7355, 72jaod 380 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M )  ->  ( A  mod  8 )  e.  S ) )
7449, 73syl5 32 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  ( A  mod  8 )  e.  S
) )
75 lgsdir2lem2.4 . . . . . 6  |-  N  e.  S
76 eleq1 2529 . . . . . 6  |-  ( ( A  mod  8 )  =  N  ->  (
( A  mod  8
)  e.  S  <->  N  e.  S ) )
7775, 76mpbiri 233 . . . . 5  |-  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S )
7877a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S
) )
7974, 78jaod 380 . . 3  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N )  ->  ( A  mod  8 )  e.  S ) )
8036, 79syl5 32 . 2  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) )
8110, 32, 803pm3.2i 1174 1  |-  ( N  e.  ZZ  /\  2  ||  ( N  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NNcn 10556   2c2 10606   4c4 10608   8c8 10612   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697    mod cmo 11999    || cdvds 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-dvds 13999
This theorem is referenced by:  lgsdir2lem3  23726
  Copyright terms: Public domain W3C validator