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Theorem 4sqlem5 13005
Description: Lemma for 4sq 13027. (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 10134 . . . 4  |-  ( ph  ->  A  e.  CC )
3 4sqlem5.4 . . . . 5  |-  B  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
41zred 10133 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
5 4sqlem5.3 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
65nnred 9777 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76rehalfcld 9974 . . . . . . . . 9  |-  ( ph  ->  ( M  /  2
)  e.  RR )
84, 7readdcld 8878 . . . . . . . 8  |-  ( ph  ->  ( A  +  ( M  /  2 ) )  e.  RR )
95nnrpd 10405 . . . . . . . 8  |-  ( ph  ->  M  e.  RR+ )
108, 9modcld 10993 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  RR )
1110recnd 8877 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  CC )
127recnd 8877 . . . . . 6  |-  ( ph  ->  ( M  /  2
)  e.  CC )
1311, 12subcld 9173 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  e.  CC )
143, 13syl5eqel 2380 . . . 4  |-  ( ph  ->  B  e.  CC )
152, 14nncand 9178 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  =  B )
162, 14subcld 9173 . . . . . 6  |-  ( ph  ->  ( A  -  B
)  e.  CC )
176recnd 8877 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185nnne0d 9806 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
1916, 17, 18divcan1d 9553 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  =  ( A  -  B ) )
203oveq2i 5885 . . . . . . . . 9  |-  ( A  -  B )  =  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )
212, 11, 12subsub3d 9203 . . . . . . . . 9  |-  ( ph  ->  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2220, 21syl5eq 2340 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2322oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  /  M
)  =  ( ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  /  2 ) )  mod  M ) )  /  M ) )
24 moddifz 10999 . . . . . . . 8  |-  ( ( ( A  +  ( M  /  2 ) )  e.  RR  /\  M  e.  RR+ )  -> 
( ( ( A  +  ( M  / 
2 ) )  -  ( ( A  +  ( M  /  2
) )  mod  M
) )  /  M
)  e.  ZZ )
258, 9, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  -  ( ( A  +  ( M  /  2
) )  mod  M
) )  /  M
)  e.  ZZ )
2623, 25eqeltrd 2370 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  M
)  e.  ZZ )
275nnzd 10132 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2826, 27zmulcld 10139 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  e.  ZZ )
2919, 28eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( A  -  B
)  e.  ZZ )
301, 29zsubcld 10138 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  e.  ZZ )
3115, 30eqeltrrd 2371 . 2  |-  ( ph  ->  B  e.  ZZ )
3231, 26jca 518 1  |-  ( ph  ->  ( B  e.  ZZ  /\  ( ( A  -  B )  /  M
)  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   RR+crp 10370    mod cmo 10989
This theorem is referenced by:  4sqlem7  13007  4sqlem8  13008  4sqlem9  13009  4sqlem10  13010  4sqlem14  13021  4sqlem15  13022  4sqlem16  13023  4sqlem17  13024  2sqlem8a  20626  2sqlem8  20627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990
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