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Theorem mulqaddmodid 10509
Description: The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
Assertion
Ref Expression
mulqaddmodid |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
Proof of Theorem mulqaddmodid
StepHypRef Expression
1 simpll 527 . . . . . 6 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> N e. ZZ )
21zcnd 9496 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> N e. CC )
3 qre 9746 . . . . . . 7 |- ( M e. QQ -> M e. RR )
43ad2antlr 489 . . . . . 6 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. RR )
54recnd 8101 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. CC )
62, 5mulcld 8093 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( N x. M ) e. CC )
7 simprr 531 . . . . . . 7 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. ( 0 [,) M ) )
8 0red 8073 . . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 e. RR )
94rexrd 8122 . . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. RR* )
10 elico2 10059 . . . . . . . 8 |- ( ( 0 e. RR /\ M e. RR* ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
118, 9, 10syl2anc 411 . . . . . . 7 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
127, 11mpbid 147 . . . . . 6 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( A e. RR /\ 0 <_ A /\ A < M ) )
1312simp1d 1012 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. RR )
1413recnd 8101 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. CC )
156, 14addcomd 8223 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( N x. M ) + A ) = ( A + ( N x. M ) ) )
1615oveq1d 5959 . 2 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( N x. M ) + A ) mod M ) = ( ( A + ( N x. M ) ) mod M ) )
17 simprl 529 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. QQ )
18 simplr 528 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. QQ )
1912simp2d 1013 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 <_ A )
2012simp3d 1014 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A < M )
218, 13, 4, 19, 20lelttrd 8197 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 < M )
22 modqcyc 10504 . . 3 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( N x. M ) ) mod M ) = ( A mod M ) )
2317, 1, 18, 21, 22syl22anc 1251 . 2 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( A + ( N x. M ) ) mod M ) = ( A mod M ) )
24 modqid 10494 . . 3 |- ( ( ( A e. QQ /\ M e. QQ ) /\ ( 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
2517, 18, 19, 20, 24syl22anc 1251 . 2 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( A mod M ) = A )
2616, 23, 253eqtrd 2242 1 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   RRcr 7924   0cc0 7925   + caddc 7928   x. cmul 7930   RR*cxr 8106   < clt 8107   <_ cle 8108   ZZcz 9372   QQcq 9740   [,)cico 10012   mod cmo 10467
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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-n0 9296  df-z 9373  df-q 9741  df-rp 9776  df-ico 10016  df-fl 10413  df-mod 10468
This theorem is referenced by:  modqmuladd  10511  addmodid  10517
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