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Theorem modaddmodlo 10555
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 10289 . . . . . . 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 9767 . . . . . 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 10511 . . . . . . . 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 9513 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. ZZ )
8 zq 9767 . . . . . 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 9776 . . . . 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 9774 . . . . 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 9515 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> B e. RR )
166nn0red 9369 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> ( A mod M ) e. RR )
17 elfzole1 10298 . . . . . 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 9371 . . . . 5 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( A mod M ) )
2015, 16, 18, 19addge0d 8615 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 <_ ( B + ( A mod M ) ) )
21 elfzolt2 10299 . . . . . 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 9069 . . . . . 6 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> M e. RR )
2415, 16, 23ltaddsubd 8638 . . . . 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 10516 . . . 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 1251 . . 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 9767 . . . . 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 9100 . . . 4 |- ( ( ( A e. ZZ /\ M e. NN ) /\ B e. ( 0..^ ( M - ( A mod M ) ) ) ) -> 0 < M )
31 modqadd2mod 10541 . . . 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 1251 . . 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 2241 . 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 1373   e. wcel 2177   class class class wbr 4051  (class class class)co 5957   0cc0 7945   + caddc 7948   < clt 8127   <_ cle 8128   - cmin 8263   NNcn 9056   NN0cn0 9315   ZZcz 9392   QQcq 9760  ..^cfzo 10284   mod cmo 10489
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490
This theorem is referenced by: (None)
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