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Theorem lgsdir2lem2 20563
Description: Lemma for lgsdir2 20567. (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 964 . . . . . 6  |-  K  e.  ZZ
5 peano2z 10060 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
64, 5ax-mp 8 . . . . 5  |-  ( K  +  1 )  e.  ZZ
72, 6eqeltri 2353 . . . 4  |-  M  e.  ZZ
8 peano2z 10060 . . . 4  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
97, 8ax-mp 8 . . 3  |-  ( M  +  1 )  e.  ZZ
101, 9eqeltri 2353 . 2  |-  N  e.  ZZ
113simp2i 965 . . . 4  |-  2  ||  ( K  +  1 )
12 2z 10054 . . . . 5  |-  2  e.  ZZ
13 dvdsadd 12567 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( 2  ||  ( K  +  1
)  <->  2  ||  (
2  +  ( K  +  1 ) ) ) )
1412, 6, 13mp2an 653 . . . 4  |-  ( 2 
||  ( K  + 
1 )  <->  2  ||  ( 2  +  ( K  +  1 ) ) )
1511, 14mpbi 199 . . 3  |-  2  ||  ( 2  +  ( K  +  1 ) )
16 zcn 10029 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
174, 16ax-mp 8 . . . . . . . . . 10  |-  K  e.  CC
18 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
1917, 18addcomi 9003 . . . . . . . . 9  |-  ( K  +  1 )  =  ( 1  +  K
)
202, 19eqtri 2303 . . . . . . . 8  |-  M  =  ( 1  +  K
)
2120oveq1i 5868 . . . . . . 7  |-  ( M  +  1 )  =  ( ( 1  +  K )  +  1 )
221, 21eqtri 2303 . . . . . 6  |-  N  =  ( ( 1  +  K )  +  1 )
23 df-2 9804 . . . . . . . 8  |-  2  =  ( 1  +  1 )
2423oveq1i 5868 . . . . . . 7  |-  ( 2  +  K )  =  ( ( 1  +  1 )  +  K
)
2518, 17, 18add32i 9030 . . . . . . 7  |-  ( ( 1  +  K )  +  1 )  =  ( ( 1  +  1 )  +  K
)
2624, 25eqtr4i 2306 . . . . . 6  |-  ( 2  +  K )  =  ( ( 1  +  K )  +  1 )
2722, 26eqtr4i 2306 . . . . 5  |-  N  =  ( 2  +  K
)
2827oveq1i 5868 . . . 4  |-  ( N  +  1 )  =  ( ( 2  +  K )  +  1 )
29 2cn 9816 . . . . 5  |-  2  e.  CC
3029, 17, 18addassi 8845 . . . 4  |-  ( ( 2  +  K )  +  1 )  =  ( 2  +  ( K  +  1 ) )
3128, 30eqtri 2303 . . 3  |-  ( N  +  1 )  =  ( 2  +  ( K  +  1 ) )
3215, 31breqtrri 4048 . 2  |-  2  ||  ( N  +  1 )
33 elfzuz2 10801 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
34 fzm1 10862 . . . . 5  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  <->  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N ) ) )
3533, 34syl 15 . . . 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 232 . . 3  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  (
( A  mod  8
)  e.  ( 0 ... ( N  - 
1 ) )  \/  ( A  mod  8
)  =  N ) )
37 elfzuz2 10801 . . . . . . . 8  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  M  e.  ( ZZ>= `  0 )
)
38 fzm1 10862 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( ( A  mod  8 )  e.  ( 0 ... M
)  <->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M ) ) )
3937, 38syl 15 . . . . . . 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 232 . . . . . 6  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
41 zcn 10029 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
4210, 41ax-mp 8 . . . . . . . 8  |-  N  e.  CC
43 zcn 10029 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
447, 43ax-mp 8 . . . . . . . 8  |-  M  e.  CC
4518, 44addcomi 9003 . . . . . . . . 9  |-  ( 1  +  M )  =  ( M  +  1 )
4645, 1eqtr4i 2306 . . . . . . . 8  |-  ( 1  +  M )  =  N
4742, 18, 44, 46subaddrii 9135 . . . . . . 7  |-  ( N  -  1 )  =  M
4847oveq2i 5869 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... M
)
4940, 48eleq2s 2375 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
5020eqcomi 2287 . . . . . . . . . 10  |-  ( 1  +  K )  =  M
5144, 18, 17, 50subaddrii 9135 . . . . . . . . 9  |-  ( M  -  1 )  =  K
5251oveq2i 5869 . . . . . . . 8  |-  ( 0 ... ( M  - 
1 ) )  =  ( 0 ... K
)
5352eleq2i 2347 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  <->  ( A  mod  8 )  e.  ( 0 ... K ) )
543simp3i 966 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... K
)  ->  ( A  mod  8 )  e.  S
) )
5553, 54syl5bi 208 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  ->  ( A  mod  8 )  e.  S
) )
56 2nn 9877 . . . . . . . . . . 11  |-  2  e.  NN
57 8nn 9883 . . . . . . . . . . 11  |-  8  e.  NN
58 4nn 9879 . . . . . . . . . . . . . . 15  |-  4  e.  NN
5958nnzi 10047 . . . . . . . . . . . . . 14  |-  4  e.  ZZ
60 dvdsmul2 12551 . . . . . . . . . . . . . 14  |-  ( ( 4  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( 4  x.  2 ) )
6159, 12, 60mp2an 653 . . . . . . . . . . . . 13  |-  2  ||  ( 4  x.  2 )
62 4t2e8 9874 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6361, 62breqtri 4046 . . . . . . . . . . . 12  |-  2  ||  8
64 dvdsmod 12585 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  /\  2  ||  8 )  ->  ( 2  ||  ( A  mod  8
)  <->  2  ||  A
) )
6563, 64mpan2 652 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  8  e.  NN  /\  A  e.  ZZ )  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6656, 57, 65mp3an12 1267 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  (
2  ||  ( A  mod  8 )  <->  2  ||  A ) )
6766notbid 285 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( -.  2  ||  ( A  mod  8 )  <->  -.  2  ||  A ) )
6867biimpar 471 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  2  ||  ( A  mod  8
) )
6911, 2breqtrri 4048 . . . . . . . . 9  |-  2  ||  M
70 id 19 . . . . . . . . 9  |-  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  =  M )
7169, 70syl5breqr 4059 . . . . . . . 8  |-  ( ( A  mod  8 )  =  M  ->  2  ||  ( A  mod  8
) )
7268, 71nsyl 113 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  -.  ( A  mod  8 )  =  M )
7372pm2.21d 98 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  e.  S
) )
7455, 73jaod 369 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  \/  ( A  mod  8 )  =  M )  ->  ( A  mod  8 )  e.  S ) )
7549, 74syl5 28 . . . 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 2343 . . . . . 6  |-  ( ( A  mod  8 )  =  N  ->  (
( A  mod  8
)  e.  S  <->  N  e.  S ) )
7876, 77mpbiri 224 . . . . 5  |-  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S )
7978a1i 10 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  =  N  ->  ( A  mod  8 )  e.  S
) )
8075, 79jaod 369 . . 3  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  \/  ( A  mod  8 )  =  N )  ->  ( A  mod  8 )  e.  S ) )
8136, 80syl5 28 . 2  |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ... N
)  ->  ( A  mod  8 )  e.  S
) )
8210, 32, 813pm3.2i 1130 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   4c4 9797   8c8 9801   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    mod cmo 10973    || cdivides 12531
This theorem is referenced by:  lgsdir2lem3  20564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-dvds 12532
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