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Theorem lgsdir2lem2 23464
Description: Lemma for lgsdir2 23468. (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 1004 . . . . . 6  |-  K  e.  ZZ
5 peano2z 10906 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
64, 5ax-mp 5 . . . . 5  |-  ( K  +  1 )  e.  ZZ
72, 6eqeltri 2525 . . . 4  |-  M  e.  ZZ
8 peano2z 10906 . . . 4  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
97, 8ax-mp 5 . . 3  |-  ( M  +  1 )  e.  ZZ
101, 9eqeltri 2525 . 2  |-  N  e.  ZZ
113simp2i 1005 . . . 4  |-  2  ||  ( K  +  1 )
12 2z 10897 . . . . 5  |-  2  e.  ZZ
13 dvdsadd 13896 . . . . 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 10870 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
174, 16ax-mp 5 . . . . . . . . . 10  |-  K  e.  CC
18 ax-1cn 9548 . . . . . . . . . 10  |-  1  e.  CC
1917, 18addcomi 9769 . . . . . . . . 9  |-  ( K  +  1 )  =  ( 1  +  K
)
202, 19eqtri 2470 . . . . . . . 8  |-  M  =  ( 1  +  K
)
2120oveq1i 6287 . . . . . . 7  |-  ( M  +  1 )  =  ( ( 1  +  K )  +  1 )
221, 21eqtri 2470 . . . . . 6  |-  N  =  ( ( 1  +  K )  +  1 )
23 df-2 10595 . . . . . . . 8  |-  2  =  ( 1  +  1 )
2423oveq1i 6287 . . . . . . 7  |-  ( 2  +  K )  =  ( ( 1  +  1 )  +  K
)
2518, 17, 18add32i 9797 . . . . . . 7  |-  ( ( 1  +  K )  +  1 )  =  ( ( 1  +  1 )  +  K
)
2624, 25eqtr4i 2473 . . . . . 6  |-  ( 2  +  K )  =  ( ( 1  +  K )  +  1 )
2722, 26eqtr4i 2473 . . . . 5  |-  N  =  ( 2  +  K
)
2827oveq1i 6287 . . . 4  |-  ( N  +  1 )  =  ( ( 2  +  K )  +  1 )
29 2cn 10607 . . . . 5  |-  2  e.  CC
3029, 17, 18addassi 9602 . . . 4  |-  ( ( 2  +  K )  +  1 )  =  ( 2  +  ( K  +  1 ) )
3128, 30eqtri 2470 . . 3  |-  ( N  +  1 )  =  ( 2  +  ( K  +  1 ) )
3215, 31breqtrri 4458 . 2  |-  2  ||  ( N  +  1 )
33 elfzuz2 11695 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
34 fzm1 11762 . . . . 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 11695 . . . . . . . 8  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  M  e.  ( ZZ>= `  0 )
)
38 fzm1 11762 . . . . . . . 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 10870 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
4210, 41ax-mp 5 . . . . . . . 8  |-  N  e.  CC
43 zcn 10870 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
447, 43ax-mp 5 . . . . . . . 8  |-  M  e.  CC
4518, 44addcomi 9769 . . . . . . . . 9  |-  ( 1  +  M )  =  ( M  +  1 )
4645, 1eqtr4i 2473 . . . . . . . 8  |-  ( 1  +  M )  =  N
4742, 18, 44, 46subaddrii 9909 . . . . . . 7  |-  ( N  -  1 )  =  M
4847oveq2i 6288 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... M
)
4940, 48eleq2s 2549 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
5020eqcomi 2454 . . . . . . . . . 10  |-  ( 1  +  K )  =  M
5144, 18, 17, 50subaddrii 9909 . . . . . . . . 9  |-  ( M  -  1 )  =  K
5251oveq2i 6288 . . . . . . . 8  |-  ( 0 ... ( M  - 
1 ) )  =  ( 0 ... K
)
5352eleq2i 2519 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  <->  ( A  mod  8 )  e.  ( 0 ... K ) )
543simp3i 1006 . . . . . . 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 10694 . . . . . . . . . . 11  |-  2  e.  NN
57 8nn 10700 . . . . . . . . . . 11  |-  8  e.  NN
58 4nn 10696 . . . . . . . . . . . . . . 15  |-  4  e.  NN
5958nnzi 10889 . . . . . . . . . . . . . 14  |-  4  e.  ZZ
60 dvdsmul2 13878 . . . . . . . . . . . . . 14  |-  ( ( 4  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( 4  x.  2 ) )
6159, 12, 60mp2an 672 . . . . . . . . . . . . 13  |-  2  ||  ( 4  x.  2 )
62 4t2e8 10690 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6361, 62breqtri 4456 . . . . . . . . . . . 12  |-  2  ||  8
64 dvdsmod 13915 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  /\  2  ||  8 )  ->  ( 2  ||  ( A  mod  8
)  <->  2  ||  A
) )
6563, 64mpan2 671 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6656, 57, 65mp3an12 1313 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6766notbid 294 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( -.  2  ||  ( A  mod  8 )  <->  -.  2  ||  A ) )
6867biimpar 485 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  2  ||  ( A  mod  8
) )
6911, 2breqtrri 4458 . . . . . . . . 9  |-  2  ||  M
70 id 22 . . . . . . . . 9  |-  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  =  M )
7169, 70syl5breqr 4469 . . . . . . . 8  |-  ( ( A  mod  8 )  =  M  ->  2  ||  ( A  mod  8
) )
7268, 71nsyl 121 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  ( A  mod  8 )  =  M )
7372pm2.21d 106 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  e.  S
) )
7455, 73jaod 380 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M )  ->  ( A  mod  8 )  e.  S ) )
7549, 74syl5 32 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  ( A  mod  8 )  e.  S
) )
76 lgsdir2lem2.4 . . . . . 6  |-  N  e.  S
77 eleq1 2513 . . . . . 6  |-  ( ( A  mod  8 )  =  N  ->  (
( A  mod  8
)  e.  S  <->  N  e.  S ) )
7876, 77mpbiri 233 . . . . 5  |-  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S )
7978a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S
) )
8075, 79jaod 380 . . 3  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N )  ->  ( A  mod  8 )  e.  S ) )
8136, 80syl5 32 . 2  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) )
8210, 32, 813pm3.2i 1173 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 972    = wceq 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    - cmin 9805   NNcn 10537   2c2 10586   4c4 10588   8c8 10592   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676    mod cmo 11970    || cdvds 13858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fl 11903  df-mod 11971  df-dvds 13859
This theorem is referenced by:  lgsdir2lem3  23465
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