ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  modaddmodup Unicode version
Theorem modaddmodup 10773
Description: The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
Assertion
Ref Expression
modaddmodup |- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( ( M - ( A mod M ) )..^ M ) -> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) ) )
Proof of Theorem modaddmodup
StepHypRef Expression
1 elfzoelz 10503 . . . . . . 7 |- ( B e. ( ( M - ( A mod M ) )..^ M ) -> B e. ZZ )
21adantr 276 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> B e. ZZ )
3 zmodcl 10730 . . . . . . . 8 |- ( ( A e. ZZ /\ M e. NN ) -> ( A mod M ) e. NN0 )
43adantl 277 . . . . . . 7 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( A mod M ) e. NN0 )
54nn0zd 9716 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( A mod M ) e. ZZ )
62, 5zaddcld 9722 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( B + ( A mod M ) ) e. ZZ )
7 zq 9976 . . . . 5 |- ( ( B + ( A mod M ) ) e. ZZ -> ( B + ( A mod M ) ) e. QQ )
86, 7syl 14 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( B + ( A mod M ) ) e. QQ )
9 simprr 533 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> M e. NN )
10 nnq 9983 . . . . 5 |- ( M e. NN -> M e. QQ )
119, 10syl 14 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> M e. QQ )
129nngt0d 9298 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> 0 < M )
13 elfzole1 10512 . . . . . 6 |- ( B e. ( ( M - ( A mod M ) )..^ M ) -> ( M - ( A mod M ) ) <_ B )
1413adantr 276 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( M - ( A mod M ) ) <_ B )
159nnred 9267 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> M e. RR )
163nn0red 9571 . . . . . . 7 |- ( ( A e. ZZ /\ M e. NN ) -> ( A mod M ) e. RR )
1716adantl 277 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( A mod M ) e. RR )
181zred 9718 . . . . . . 7 |- ( B e. ( ( M - ( A mod M ) )..^ M ) -> B e. RR )
1918adantr 276 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> B e. RR )
2015, 17, 19lesubaddd 8833 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( ( M - ( A mod M ) ) <_ B <-> M <_ ( B + ( A mod M ) ) ) )
2114, 20mpbid 147 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> M <_ ( B + ( A mod M ) ) )
22 elfzolt2 10513 . . . . . . 7 |- ( B e. ( ( M - ( A mod M ) )..^ M ) -> B < M )
2322adantr 276 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> B < M )
24 zq 9976 . . . . . . . 8 |- ( A e. ZZ -> A e. QQ )
2524ad2antrl 490 . . . . . . 7 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> A e. QQ )
26 modqlt 10719 . . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) < M )
2725, 11, 12, 26syl3anc 1274 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( A mod M ) < M )
2819, 17, 15, 15, 23, 27lt2addd 8858 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( B + ( A mod M ) ) < ( M + M ) )
299nncnd 9268 . . . . . 6 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> M e. CC )
30292timesd 9498 . . . . 5 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( 2 x. M ) = ( M + M ) )
3128, 30breqtrrd 4142 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( B + ( A mod M ) ) < ( 2 x. M ) )
32 q2submod 10771 . . . 4 |- ( ( ( ( B + ( A mod M ) ) e. QQ /\ M e. QQ /\ 0 < M ) /\ ( M <_ ( B + ( A mod M ) ) /\ ( B + ( A mod M ) ) < ( 2 x. M ) ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + ( A mod M ) ) - M ) )
338, 11, 12, 21, 31, 32syl32anc 1282 . . 3 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + ( A mod M ) ) - M ) )
34 zq 9976 . . . . 5 |- ( B e. ZZ -> B e. QQ )
352, 34syl 14 . . . 4 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> B e. QQ )
36 modqadd2mod 10760 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
3725, 35, 11, 12, 36syl22anc 1275 . . 3 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
3833, 37eqtr3d 2269 . 2 |- ( ( B e. ( ( M - ( A mod M ) )..^ M ) /\ ( A e. ZZ /\ M e. NN ) ) -> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) )
3938expcom 116 1 |- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( ( M - ( A mod M ) )..^ M ) -> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   QQcq 9969  ..^cfzo 10498   mod cmo 10708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator