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Theorem 4sqlem5 12989
Description: Lemma for 4sq 13011. (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 10118 . . . 4  |-  ( ph  ->  A  e.  CC )
3 4sqlem5.4 . . . . 5  |-  B  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
41zred 10117 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
5 4sqlem5.3 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
65nnred 9761 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76rehalfcld 9958 . . . . . . . . 9  |-  ( ph  ->  ( M  /  2
)  e.  RR )
84, 7readdcld 8862 . . . . . . . 8  |-  ( ph  ->  ( A  +  ( M  /  2 ) )  e.  RR )
95nnrpd 10389 . . . . . . . 8  |-  ( ph  ->  M  e.  RR+ )
108, 9modcld 10977 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  RR )
1110recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( M  /  2
) )  mod  M
)  e.  CC )
127recnd 8861 . . . . . 6  |-  ( ph  ->  ( M  /  2
)  e.  CC )
1311, 12subcld 9157 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  e.  CC )
143, 13syl5eqel 2367 . . . 4  |-  ( ph  ->  B  e.  CC )
152, 14nncand 9162 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  =  B )
162, 14subcld 9157 . . . . . 6  |-  ( ph  ->  ( A  -  B
)  e.  CC )
176recnd 8861 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185nnne0d 9790 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
1916, 17, 18divcan1d 9537 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  =  ( A  -  B ) )
203oveq2i 5869 . . . . . . . . 9  |-  ( A  -  B )  =  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )
212, 11, 12subsub3d 9187 . . . . . . . . 9  |-  ( ph  ->  ( A  -  (
( ( A  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) ) )  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2220, 21syl5eq 2327 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  =  ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  / 
2 ) )  mod 
M ) ) )
2322oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  /  M
)  =  ( ( ( A  +  ( M  /  2 ) )  -  ( ( A  +  ( M  /  2 ) )  mod  M ) )  /  M ) )
24 moddifz 10983 . . . . . . . 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 2357 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  M
)  e.  ZZ )
275nnzd 10116 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2826, 27zmulcld 10123 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B )  /  M )  x.  M
)  e.  ZZ )
2919, 28eqeltrrd 2358 . . . 4  |-  ( ph  ->  ( A  -  B
)  e.  ZZ )
301, 29zsubcld 10122 . . 3  |-  ( ph  ->  ( A  -  ( A  -  B )
)  e.  ZZ )
3115, 30eqeltrrd 2358 . 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 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   RRcr 8736    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   RR+crp 10354    mod cmo 10973
This theorem is referenced by:  4sqlem7  12991  4sqlem8  12992  4sqlem9  12993  4sqlem10  12994  4sqlem14  13005  4sqlem15  13006  4sqlem16  13007  4sqlem17  13008  2sqlem8a  20610  2sqlem8  20611
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-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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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