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Theorem 4sqlem5 12321
Description: Lemma for 4sq (not yet proved here). (Contributed by Mario Carneiro, 15-Jul-2014.)
Hypotheses
Ref Expression
4sqlem5.2  |-  ( ph  ->  A  e.  ZZ )
4sqlem5.3  |-  ( ph  ->  M  e.  NN )
4sqlem5.4  |-  B  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
Assertion
Ref Expression
4sqlem5  |-  ( ph  ->  ( B  e.  ZZ  /\  ( ( A  -  B )  /  M
)  e.  ZZ ) )

Proof of Theorem 4sqlem5
StepHypRef Expression
1 4sqlem5.2 . . . . 5  |-  ( ph  ->  A  e.  ZZ )
21zcnd 9322 . . . 4  |-  ( ph  ->  A  e.  CC )
3 4sqlem5.4 . . . . 5  |-  B  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
4 zq 9572 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  QQ )
51, 4syl 14 . . . . . . . . 9  |-  ( ph  ->  A  e.  QQ )
6 4sqlem5.3 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
76nnzd 9320 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
8 2nn 9026 . . . . . . . . . 10  |-  2  e.  NN
9 znq 9570 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  2  e.  NN )  ->  ( M  /  2
)  e.  QQ )
107, 8, 9sylancl 411 . . . . . . . . 9  |-  ( ph  ->  ( M  /  2
)  e.  QQ )
11 qaddcl 9581 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( M  /  2
)  e.  QQ )  ->  ( A  +  ( M  /  2
) )  e.  QQ )
125, 10, 11syl2anc 409 . . . . . . . 8  |-  ( ph  ->  ( A  +  ( M  /  2 ) )  e.  QQ )
13 nnq 9579 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  QQ )
146, 13syl 14 . . . . . . . 8  |-  ( ph  ->  M  e.  QQ )
156nngt0d 8909 . . . . . . . 8  |-  ( ph  ->  0  <  M )
1612, 14, 15modqcld 10271 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  QQ )
17 qcn 9580 . . . . . . 7  |-  ( ( ( A  +  ( M  /  2 ) )  mod  M )  e.  QQ  ->  (
( A  +  ( M  /  2 ) )  mod  M )  e.  CC )
1816, 17syl 14 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  CC )
196nnred 8878 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
2019rehalfcld 9111 . . . . . . 7  |-  ( ph  ->  ( M  /  2
)  e.  RR )
2120recnd 7935 . . . . . 6  |-  ( ph  ->  ( M  /  2
)  e.  CC )
2218, 21subcld 8217 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  e.  CC )
233, 22eqeltrid 2257 . . . 4  |-  ( ph  ->  B  e.  CC )
242, 23nncand 8222 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  =  B )
252, 23subcld 8217 . . . . . 6  |-  ( ph  ->  ( A  -  B
)  e.  CC )
2619recnd 7935 . . . . . 6  |-  ( ph  ->  M  e.  CC )
276nnap0d 8911 . . . . . 6  |-  ( ph  ->  M #  0 )
2825, 26, 27divcanap1d 8695 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  =  ( A  -  B ) )
293oveq2i 5861 . . . . . . . . 9  |-  ( A  -  B )  =  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )
302, 18, 21subsub3d 8247 . . . . . . . . 9  |-  ( ph  ->  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
3129, 30eqtrid 2215 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
3231oveq1d 5865 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  /  M
)  =  ( ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  /  2 ) )  mod  M ) )  /  M ) )
33 modqdifz 10279 . . . . . . . 8  |-  ( ( ( A  +  ( M  /  2 ) )  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( ( A  +  ( M  /  2
) )  -  (
( A  +  ( M  /  2 ) )  mod  M ) )  /  M )  e.  ZZ )
3412, 14, 15, 33syl3anc 1233 . . . . . . 7  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  -  ( ( A  +  ( M  /  2
) )  mod  M
) )  /  M
)  e.  ZZ )
3532, 34eqeltrd 2247 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  M
)  e.  ZZ )
3635, 7zmulcld 9327 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  e.  ZZ )
3728, 36eqeltrrd 2248 . . . 4  |-  ( ph  ->  ( A  -  B
)  e.  ZZ )
381, 37zsubcld 9326 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  e.  ZZ )
3924, 38eqeltrrd 2248 . 2  |-  ( ph  ->  B  e.  ZZ )
4039, 35jca 304 1  |-  ( ph  ->  ( B  e.  ZZ  /\  ( ( A  -  B )  /  M
)  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761    + caddc 7764    x. cmul 7766    < clt 7941    - cmin 8077    / cdiv 8576   NNcn 8865   2c2 8916   ZZcz 9199   QQcq 9565    mod cmo 10265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-n0 9123  df-z 9200  df-q 9566  df-rp 9598  df-fl 10213  df-mod 10266
This theorem is referenced by:  4sqlem7  12323  4sqlem8  12324  4sqlem9  12325  4sqlem10  12326  2sqlem8a  13673  2sqlem8  13674
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