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Theorem lgsdir2lem2 20976
Description: Lemma for lgsdir2 20980. (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 966 . . . . . 6  |-  K  e.  ZZ
5 peano2z 10251 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
64, 5ax-mp 8 . . . . 5  |-  ( K  +  1 )  e.  ZZ
72, 6eqeltri 2458 . . . 4  |-  M  e.  ZZ
8 peano2z 10251 . . . 4  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
97, 8ax-mp 8 . . 3  |-  ( M  +  1 )  e.  ZZ
101, 9eqeltri 2458 . 2  |-  N  e.  ZZ
113simp2i 967 . . . 4  |-  2  ||  ( K  +  1 )
12 2z 10245 . . . . 5  |-  2  e.  ZZ
13 dvdsadd 12816 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( 2  ||  ( K  +  1
)  <->  2  ||  (
2  +  ( K  +  1 ) ) ) )
1412, 6, 13mp2an 654 . . . 4  |-  ( 2 
||  ( K  + 
1 )  <->  2  ||  ( 2  +  ( K  +  1 ) ) )
1511, 14mpbi 200 . . 3  |-  2  ||  ( 2  +  ( K  +  1 ) )
16 zcn 10220 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
174, 16ax-mp 8 . . . . . . . . . 10  |-  K  e.  CC
18 ax-1cn 8982 . . . . . . . . . 10  |-  1  e.  CC
1917, 18addcomi 9190 . . . . . . . . 9  |-  ( K  +  1 )  =  ( 1  +  K
)
202, 19eqtri 2408 . . . . . . . 8  |-  M  =  ( 1  +  K
)
2120oveq1i 6031 . . . . . . 7  |-  ( M  +  1 )  =  ( ( 1  +  K )  +  1 )
221, 21eqtri 2408 . . . . . 6  |-  N  =  ( ( 1  +  K )  +  1 )
23 df-2 9991 . . . . . . . 8  |-  2  =  ( 1  +  1 )
2423oveq1i 6031 . . . . . . 7  |-  ( 2  +  K )  =  ( ( 1  +  1 )  +  K
)
2518, 17, 18add32i 9217 . . . . . . 7  |-  ( ( 1  +  K )  +  1 )  =  ( ( 1  +  1 )  +  K
)
2624, 25eqtr4i 2411 . . . . . 6  |-  ( 2  +  K )  =  ( ( 1  +  K )  +  1 )
2722, 26eqtr4i 2411 . . . . 5  |-  N  =  ( 2  +  K
)
2827oveq1i 6031 . . . 4  |-  ( N  +  1 )  =  ( ( 2  +  K )  +  1 )
29 2cn 10003 . . . . 5  |-  2  e.  CC
3029, 17, 18addassi 9032 . . . 4  |-  ( ( 2  +  K )  +  1 )  =  ( 2  +  ( K  +  1 ) )
3128, 30eqtri 2408 . . 3  |-  ( N  +  1 )  =  ( 2  +  ( K  +  1 ) )
3215, 31breqtrri 4179 . 2  |-  2  ||  ( N  +  1 )
33 elfzuz2 10995 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
34 fzm1 11058 . . . . 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 233 . . 3  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  (
( A  mod  8
)  e.  ( 0 ... ( N  - 
1 ) )  \/  ( A  mod  8
)  =  N ) )
37 elfzuz2 10995 . . . . . . . 8  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  M  e.  ( ZZ>= `  0 )
)
38 fzm1 11058 . . . . . . . 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 233 . . . . . 6  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
41 zcn 10220 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
4210, 41ax-mp 8 . . . . . . . 8  |-  N  e.  CC
43 zcn 10220 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
447, 43ax-mp 8 . . . . . . . 8  |-  M  e.  CC
4518, 44addcomi 9190 . . . . . . . . 9  |-  ( 1  +  M )  =  ( M  +  1 )
4645, 1eqtr4i 2411 . . . . . . . 8  |-  ( 1  +  M )  =  N
4742, 18, 44, 46subaddrii 9322 . . . . . . 7  |-  ( N  -  1 )  =  M
4847oveq2i 6032 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... M
)
4940, 48eleq2s 2480 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
5020eqcomi 2392 . . . . . . . . . 10  |-  ( 1  +  K )  =  M
5144, 18, 17, 50subaddrii 9322 . . . . . . . . 9  |-  ( M  -  1 )  =  K
5251oveq2i 6032 . . . . . . . 8  |-  ( 0 ... ( M  - 
1 ) )  =  ( 0 ... K
)
5352eleq2i 2452 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  <->  ( A  mod  8 )  e.  ( 0 ... K ) )
543simp3i 968 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... K
)  ->  ( A  mod  8 )  e.  S
) )
5553, 54syl5bi 209 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  ->  ( A  mod  8 )  e.  S
) )
56 2nn 10066 . . . . . . . . . . 11  |-  2  e.  NN
57 8nn 10072 . . . . . . . . . . 11  |-  8  e.  NN
58 4nn 10068 . . . . . . . . . . . . . . 15  |-  4  e.  NN
5958nnzi 10238 . . . . . . . . . . . . . 14  |-  4  e.  ZZ
60 dvdsmul2 12800 . . . . . . . . . . . . . 14  |-  ( ( 4  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( 4  x.  2 ) )
6159, 12, 60mp2an 654 . . . . . . . . . . . . 13  |-  2  ||  ( 4  x.  2 )
62 4t2e8 10063 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6361, 62breqtri 4177 . . . . . . . . . . . 12  |-  2  ||  8
64 dvdsmod 12834 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  /\  2  ||  8 )  ->  ( 2  ||  ( A  mod  8
)  <->  2  ||  A
) )
6563, 64mpan2 653 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6656, 57, 65mp3an12 1269 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6766notbid 286 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( -.  2  ||  ( A  mod  8 )  <->  -.  2  ||  A ) )
6867biimpar 472 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  2  ||  ( A  mod  8
) )
6911, 2breqtrri 4179 . . . . . . . . 9  |-  2  ||  M
70 id 20 . . . . . . . . 9  |-  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  =  M )
7169, 70syl5breqr 4190 . . . . . . . 8  |-  ( ( A  mod  8 )  =  M  ->  2  ||  ( A  mod  8
) )
7268, 71nsyl 115 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  ( A  mod  8 )  =  M )
7372pm2.21d 100 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  e.  S
) )
7455, 73jaod 370 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M )  ->  ( A  mod  8 )  e.  S ) )
7549, 74syl5 30 . . . 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 2448 . . . . . 6  |-  ( ( A  mod  8 )  =  N  ->  (
( A  mod  8
)  e.  S  <->  N  e.  S ) )
7876, 77mpbiri 225 . . . . 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 370 . . 3  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N )  ->  ( A  mod  8 )  e.  S ) )
8136, 80syl5 30 . 2  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) )
8210, 32, 813pm3.2i 1132 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    - cmin 9224   NNcn 9933   2c2 9982   4c4 9984   8c8 9988   ZZcz 10215   ZZ>=cuz 10421   ...cfz 10976    mod cmo 11178    || cdivides 12780
This theorem is referenced by:  lgsdir2lem3  20977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-fl 11130  df-mod 11179  df-dvds 12781
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