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Theorem modaddmodlo 10531
Description: The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
Assertion
Ref Expression
modaddmodlo |- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) ) )
Proof of Theorem modaddmodlo
StepHypRef Expression
1 elfzoelz 10268 . . . . . . 7 |- ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> B e. ZZ )
21adantl 277 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. ZZ )
3 zq 9746 . . . . . 6 |- ( B e. ZZ -> B e. QQ )
42, 3syl 14 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. QQ )
5 zmodcl 10487 . . . . . . . 8 |- ( ( A e. ZZ /\ M e. NN ) -> ( A mod M ) e. NN0 )
65adantr 276 . . . . . . 7 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. NN0 )
76nn0zd 9492 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. ZZ )
8 zq 9746 . . . . . 6 |- ( ( A mod M ) e. ZZ -> ( A mod M ) e. QQ )
97, 8syl 14 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. QQ )
10 qaddcl 9755 . . . . 5 |- ( ( B e. QQ /\ ( A mod M ) e. QQ ) -> ( B + ( A mod M ) ) e. QQ )
114, 9, 10syl2anc 411 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( B + ( A mod M ) ) e. QQ )
12 simplr 528 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. NN )
13 nnq 9753 . . . . 5 |- ( M e. NN -> M e. QQ )
1412, 13syl 14 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. QQ )
152zred 9494 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. RR )
166nn0red 9348 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. RR )
17 elfzole1 10277 . . . . . 6 |- ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> 0 <_ B )
1817adantl 277 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ B )
196nn0ge0d 9350 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( A mod M ) )
2015, 16, 18, 19addge0d 8594 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( B + ( A mod M ) ) )
21 elfzolt2 10278 . . . . . 6 |- ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> B < ( M - ( A mod M ) ) )
2221adantl 277 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B < ( M - ( A mod M ) ) )
2312nnred 9048 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. RR )
2415, 16, 23ltaddsubd 8617 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( ( B + ( A mod M ) ) < M <-> B < ( M - ( A mod M ) ) ) )
2522, 24mpbird 167 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( B + ( A mod M ) ) < M )
26 modqid 10492 . . . 4 |- ( ( ( ( B + ( A mod M ) ) e. QQ /\ M e. QQ ) /\ ( 0 <_ ( B + ( A mod M ) ) /\ ( B + ( A mod M ) ) < M ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( B + ( A mod M ) ) )
2711, 14, 20, 25, 26syl22anc 1250 . . 3 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( B + ( A mod M ) ) )
28 zq 9746 . . . . 5 |- ( A e. ZZ -> A e. QQ )
2928ad2antrr 488 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> A e. QQ )
3012nngt0d 9079 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 < M )
31 modqadd2mod 10517 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
3229, 4, 14, 30, 31syl22anc 1250 . . 3 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
3327, 32eqtr3d 2239 . 2 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) )
3433ex 115 1 |- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5943   0cc0 7924   + caddc 7927   < clt 8106   <_ cle 8107   - cmin 8242   NNcn 9035   NN0cn0 9294   ZZcz 9371   QQcq 9739  ..^cfzo 10263   mod cmo 10465
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-po 4342  df-iso 4343  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466
This theorem is referenced by: (None)
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