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Theorem modaddmodlo 10825
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 10554 . . . . . . 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 10026 . . . . . 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 10781 . . . . . . . 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 9766 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. ZZ )
8 zq 10026 . . . . . 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 10035 . . . . 5 |- ( ( B e. QQ /\ ( A mod M ) e. QQ ) -> ( B + ( A mod M ) ) e. QQ )
114, 9, 10syl2anc 415 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( B + ( A mod M ) ) e. QQ )
12 simplr 533 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. NN )
13 nnq 10033 . . . . 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 9768 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. RR )
166nn0red 9621 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. RR )
17 elfzole1 10563 . . . . . 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 9623 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( A mod M ) )
2015, 16, 18, 19addge0d 8850 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( B + ( A mod M ) ) )
21 elfzolt2 10564 . . . . . 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 9317 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. RR )
2415, 16, 23ltaddsubd 8873 . . . . 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 10786 . . . 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 1279 . . 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 10026 . . . . 5 |- ( A e. ZZ -> A e. QQ )
2928ad2antrr 492 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> A e. QQ )
3012nngt0d 9348 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 < M )
31 modqadd2mod 10811 . . . 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 1279 . . 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 2273 . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   0cc0 8179   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   NNcn 9304   NN0cn0 9563   ZZcz 9644   QQcq 10019  ..^cfzo 10549   mod cmo 10759
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760
This theorem is used by: (None)
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