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Theorem lgsdir2lem2 23320
Description: Lemma for lgsdir2 23324. (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 1000 . . . . . 6  |-  K  e.  ZZ
5 peano2z 10893 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
64, 5ax-mp 5 . . . . 5  |-  ( K  +  1 )  e.  ZZ
72, 6eqeltri 2544 . . . 4  |-  M  e.  ZZ
8 peano2z 10893 . . . 4  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
97, 8ax-mp 5 . . 3  |-  ( M  +  1 )  e.  ZZ
101, 9eqeltri 2544 . 2  |-  N  e.  ZZ
113simp2i 1001 . . . 4  |-  2  ||  ( K  +  1 )
12 2z 10885 . . . . 5  |-  2  e.  ZZ
13 dvdsadd 13872 . . . . 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 10858 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
174, 16ax-mp 5 . . . . . . . . . 10  |-  K  e.  CC
18 ax-1cn 9539 . . . . . . . . . 10  |-  1  e.  CC
1917, 18addcomi 9759 . . . . . . . . 9  |-  ( K  +  1 )  =  ( 1  +  K
)
202, 19eqtri 2489 . . . . . . . 8  |-  M  =  ( 1  +  K
)
2120oveq1i 6285 . . . . . . 7  |-  ( M  +  1 )  =  ( ( 1  +  K )  +  1 )
221, 21eqtri 2489 . . . . . 6  |-  N  =  ( ( 1  +  K )  +  1 )
23 df-2 10583 . . . . . . . 8  |-  2  =  ( 1  +  1 )
2423oveq1i 6285 . . . . . . 7  |-  ( 2  +  K )  =  ( ( 1  +  1 )  +  K
)
2518, 17, 18add32i 9787 . . . . . . 7  |-  ( ( 1  +  K )  +  1 )  =  ( ( 1  +  1 )  +  K
)
2624, 25eqtr4i 2492 . . . . . 6  |-  ( 2  +  K )  =  ( ( 1  +  K )  +  1 )
2722, 26eqtr4i 2492 . . . . 5  |-  N  =  ( 2  +  K
)
2827oveq1i 6285 . . . 4  |-  ( N  +  1 )  =  ( ( 2  +  K )  +  1 )
29 2cn 10595 . . . . 5  |-  2  e.  CC
3029, 17, 18addassi 9593 . . . 4  |-  ( ( 2  +  K )  +  1 )  =  ( 2  +  ( K  +  1 ) )
3128, 30eqtri 2489 . . 3  |-  ( N  +  1 )  =  ( 2  +  ( K  +  1 ) )
3215, 31breqtrri 4465 . 2  |-  2  ||  ( N  +  1 )
33 elfzuz2 11680 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
34 fzm1 11747 . . . . 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 11680 . . . . . . . 8  |-  ( ( A  mod  8 )  e.  ( 0 ... M )  ->  M  e.  ( ZZ>= `  0 )
)
38 fzm1 11747 . . . . . . . 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 10858 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
4210, 41ax-mp 5 . . . . . . . 8  |-  N  e.  CC
43 zcn 10858 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
447, 43ax-mp 5 . . . . . . . 8  |-  M  e.  CC
4518, 44addcomi 9759 . . . . . . . . 9  |-  ( 1  +  M )  =  ( M  +  1 )
4645, 1eqtr4i 2492 . . . . . . . 8  |-  ( 1  +  M )  =  N
4742, 18, 44, 46subaddrii 9897 . . . . . . 7  |-  ( N  -  1 )  =  M
4847oveq2i 6286 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... M
)
4940, 48eleq2s 2568 . . . . 5  |-  ( ( A  mod  8 )  e.  ( 0 ... ( N  -  1 ) )  ->  (
( A  mod  8
)  e.  ( 0 ... ( M  - 
1 ) )  \/  ( A  mod  8
)  =  M ) )
5020eqcomi 2473 . . . . . . . . . 10  |-  ( 1  +  K )  =  M
5144, 18, 17, 50subaddrii 9897 . . . . . . . . 9  |-  ( M  -  1 )  =  K
5251oveq2i 6286 . . . . . . . 8  |-  ( 0 ... ( M  - 
1 ) )  =  ( 0 ... K
)
5352eleq2i 2538 . . . . . . 7  |-  ( ( A  mod  8 )  e.  ( 0 ... ( M  -  1 ) )  <->  ( A  mod  8 )  e.  ( 0 ... K ) )
543simp3i 1002 . . . . . . 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 10682 . . . . . . . . . . 11  |-  2  e.  NN
57 8nn 10688 . . . . . . . . . . 11  |-  8  e.  NN
58 4nn 10684 . . . . . . . . . . . . . . 15  |-  4  e.  NN
5958nnzi 10877 . . . . . . . . . . . . . 14  |-  4  e.  ZZ
60 dvdsmul2 13856 . . . . . . . . . . . . . 14  |-  ( ( 4  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( 4  x.  2 ) )
6159, 12, 60mp2an 672 . . . . . . . . . . . . 13  |-  2  ||  ( 4  x.  2 )
62 4t2e8 10678 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6361, 62breqtri 4463 . . . . . . . . . . . 12  |-  2  ||  8
64 dvdsmod 13891 . . . . . . . . . . . 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 1309 . . . . . . . . . 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 4465 . . . . . . . . 9  |-  2  ||  M
70 id 22 . . . . . . . . 9  |-  ( ( A  mod  8 )  =  M  ->  ( A  mod  8 )  =  M )
7169, 70syl5breqr 4476 . . . . . . . 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 2532 . . . . . 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 1169 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794   NNcn 10525   2c2 10574   4c4 10576   8c8 10580   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661    mod cmo 11952    || cdivides 13836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fl 11886  df-mod 11953  df-dvds 13837
This theorem is referenced by:  lgsdir2lem3  23321
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