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Theorem modaddmodlo 10462
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 10216 . . . . . . 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 9694 . . . . . 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 10418 . . . . . . . 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 9440 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. ZZ )
8 zq 9694 . . . . . 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 9703 . . . . 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 9701 . . . . 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 9442 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. RR )
166nn0red 9297 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. RR )
17 elfzole1 10225 . . . . . 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 9299 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( A mod M ) )
2015, 16, 18, 19addge0d 8543 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( B + ( A mod M ) ) )
21 elfzolt2 10226 . . . . . 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 8997 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. RR )
2415, 16, 23ltaddsubd 8566 . . . . 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 10423 . . . 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 9694 . . . . 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 9028 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 < M )
31 modqadd2mod 10448 . . . 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 2228 . 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 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   0cc0 7874   + caddc 7877   < clt 8056   <_ cle 8057   - cmin 8192   NNcn 8984   NN0cn0 9243   ZZcz 9320   QQcq 9687  ..^cfzo 10211   mod cmo 10396
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397
This theorem is referenced by: (None)
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