Theorem modqmulmodr 10333

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 524	. . . . 5 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. ZZ )
2	1	zcnd 9322	. . . 4 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. CC )
3		simplr 525	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
4		simprl 526	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
5		simprr 527	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
6	3, 4, 5	modqcld 10271	. . . . 5 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
7		qcn 9580	. . . . 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 7928	. . 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 5865	. 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 10332	. . 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 564	. 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 9580	. . . . 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 7928	. . 3 |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. A ) = ( A x. B ) )
16	15	oveq1d 5865	. 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 2207	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 103   = wceq 1348   e. wcel 2141   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761   x. cmul 7766   < clt 7941   ZZcz 9199   QQcq 9565   mod cmo 10265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-n0 9123  df-z 9200  df-q 9566  df-rp 9598  df-fl 10213  df-mod 10266

This theorem is referenced by: (None)