Theorem modqmulmodr 10384

Description: The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)

Assertion

Ref	Expression
modqmulmodr	|- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) )

Proof of Theorem modqmulmodr

Step	Hyp	Ref	Expression
1		simpll 527	. . . . 5 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. ZZ )
2	1	zcnd 9371	. . . 4 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. CC )
3		simplr 528	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
4		simprl 529	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
5		simprr 531	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
6	3, 4, 5	modqcld 10322	. . . . 5 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
7		qcn 9629	. . . . 5 |- ( ( B mod M ) e. QQ -> ( B mod M ) e. CC )
8	6, 7	syl 14	. . . 4 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. CC )
9	2, 8	mulcomd 7974	. . 3 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A x. ( B mod M ) ) = ( ( B mod M ) x. A ) )
10	9	oveq1d 5886	. 2 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( ( B mod M ) x. A ) mod M ) )
11		modqmulmod 10383	. . 3 |- ( ( ( B e. QQ /\ A e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. A ) mod M ) = ( ( B x. A ) mod M ) )
12	11	ancom1s 569	. 2 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. A ) mod M ) = ( ( B x. A ) mod M ) )
13		qcn 9629	. . . . 5 |- ( B e. QQ -> B e. CC )
14	3, 13	syl 14	. . . 4 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. CC )
15	14, 2	mulcomd 7974	. . 3 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. A ) = ( A x. B ) )
16	15	oveq1d 5886	. 2 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. A ) mod M ) = ( ( A x. B ) mod M ) )
17	10, 12, 16	3eqtrd 2214	1 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4002  (class class class)co 5871   CCcc 7805   0cc0 7807   x. cmul 7812   < clt 7987   ZZcz 9248   QQcq 9614   mod cmo 10316

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-po 4295  df-iso 4296  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-n0 9172  df-z 9249  df-q 9615  df-rp 9649  df-fl 10264  df-mod 10317

This theorem is referenced by: (None)