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Theorem mulqaddmodid 10822
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 531 . . . . . 6 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> N e. ZZ )
21zcnd 9780 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> N e. CC )
3 qre 10041 . . . . . . 7 |- ( M e. QQ -> M e. RR )
43ad2antlr 493 . . . . . 6 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. RR )
54recnd 8360 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. CC )
62, 5mulcld 8352 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( N x. M ) e. CC )
7 simprr 537 . . . . . . 7 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. ( 0 [,) M ) )
8 0red 8333 . . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 e. RR )
94rexrd 8381 . . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. RR* )
10 elico2 10356 . . . . . . . 8 |- ( ( 0 e. RR /\ M e. RR* ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
118, 9, 10syl2anc 415 . . . . . . 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 1040 . . . . 5 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. RR )
1413recnd 8360 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. CC )
156, 14addcomd 8484 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( N x. M ) + A ) = ( A + ( N x. M ) ) )
1615oveq1d 6103 . 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 535 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A e. QQ )
18 simplr 533 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> M e. QQ )
1912simp2d 1041 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 <_ A )
2012simp3d 1042 . . . 4 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> A < M )
218, 13, 4, 19, 20lelttrd 8458 . . 3 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> 0 < M )
22 modqcyc 10817 . . 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 1279 . 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 10807 . . 3 |- ( ( ( A e. QQ /\ M e. QQ ) /\ ( 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
2517, 18, 19, 20, 24syl22anc 1279 . 2 |- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( A mod M ) = A )
2616, 23, 253eqtrd 2275 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   RRcr 8184   0cc0 8185   + caddc 8188   x. cmul 8190   RR*cxr 8365   < clt 8366   <_ cle 8367   ZZcz 9655   QQcq 10035   [,)cico 10309   mod cmo 10780
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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304
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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-ico 10313  df-fl 10722  df-mod 10781
This theorem is used by:  modqmuladd  10824  addmodid  10830
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