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Theorem 4sqlem5 13302
Description: Lemma for 4sq 13324. (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 10368 . . . 4  |-  ( ph  ->  A  e.  CC )
3 4sqlem5.4 . . . . 5  |-  B  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
41zred 10367 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
5 4sqlem5.3 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
65nnred 10007 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76rehalfcld 10206 . . . . . . . . 9  |-  ( ph  ->  ( M  /  2
)  e.  RR )
84, 7readdcld 9107 . . . . . . . 8  |-  ( ph  ->  ( A  +  ( M  /  2 ) )  e.  RR )
95nnrpd 10639 . . . . . . . 8  |-  ( ph  ->  M  e.  RR+ )
108, 9modcld 11246 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  RR )
1110recnd 9106 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  CC )
127recnd 9106 . . . . . 6  |-  ( ph  ->  ( M  /  2
)  e.  CC )
1311, 12subcld 9403 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  e.  CC )
143, 13syl5eqel 2519 . . . 4  |-  ( ph  ->  B  e.  CC )
152, 14nncand 9408 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  =  B )
162, 14subcld 9403 . . . . . 6  |-  ( ph  ->  ( A  -  B
)  e.  CC )
176recnd 9106 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185nnne0d 10036 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
1916, 17, 18divcan1d 9783 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  =  ( A  -  B ) )
203oveq2i 6084 . . . . . . . . 9  |-  ( A  -  B )  =  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )
212, 11, 12subsub3d 9433 . . . . . . . . 9  |-  ( ph  ->  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2220, 21syl5eq 2479 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2322oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  /  M
)  =  ( ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  /  2 ) )  mod  M ) )  /  M ) )
24 moddifz 11252 . . . . . . . 8  |-  ( ( ( A  +  ( M  /  2 ) )  e.  RR  /\  M  e.  RR+ )  -> 
( ( ( A  +  ( M  / 
2 ) )  -  ( ( A  +  ( M  /  2
) )  mod  M
) )  /  M
)  e.  ZZ )
258, 9, 24syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  -  ( ( A  +  ( M  /  2
) )  mod  M
) )  /  M
)  e.  ZZ )
2623, 25eqeltrd 2509 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  M
)  e.  ZZ )
275nnzd 10366 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2826, 27zmulcld 10373 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  e.  ZZ )
2919, 28eqeltrrd 2510 . . . 4  |-  ( ph  ->  ( A  -  B
)  e.  ZZ )
301, 29zsubcld 10372 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  e.  ZZ )
3115, 30eqeltrrd 2510 . 2  |-  ( ph  ->  B  e.  ZZ )
3231, 26jca 519 1  |-  ( ph  ->  ( B  e.  ZZ  /\  ( ( A  -  B )  /  M
)  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980   RRcr 8981    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   ZZcz 10274   RR+crp 10604    mod cmo 11242
This theorem is referenced by:  4sqlem7  13304  4sqlem8  13305  4sqlem9  13306  4sqlem10  13307  4sqlem14  13318  4sqlem15  13319  4sqlem16  13320  4sqlem17  13321  2sqlem8a  21147  2sqlem8  21148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243
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