Theorem modqdi 10327

Description: Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)

Assertion

Ref	Expression
modqdi	|- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) )

Proof of Theorem modqdi

Step	Hyp	Ref	Expression
1		simp1l 1011	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2		qcn 9572	. . . . 5 |- ( A e. QQ -> A e. CC )
3	1, 2	syl 14	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. CC )
4		simp2 988	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
5		qcn 9572	. . . . 5 |- ( B e. QQ -> B e. CC )
6	4, 5	syl 14	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. CC )
7		simp3l 1015	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
8		simp3r 1016	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
9	8	gt0ne0d 8410	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C =/= 0 )
10		qdivcl 9581	. . . . . . . . 9 |- ( ( B e. QQ /\ C e. QQ /\ C =/= 0 ) -> ( B / C ) e. QQ )
11	4, 7, 9, 10	syl3anc 1228	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B / C ) e. QQ )
12	11	flqcld 10212	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. ZZ )
13		zq 9564	. . . . . . 7 |- ( ( |_ ` ( B / C ) ) e. ZZ -> ( |_ ` ( B / C ) ) e. QQ )
14	12, 13	syl 14	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. QQ )
15		qmulcl 9575	. . . . . 6 |- ( ( C e. QQ /\ ( |_ ` ( B / C ) ) e. QQ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
16	7, 14, 15	syl2anc 409	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
17		qcn 9572	. . . . 5 |- ( ( C x. ( |_ ` ( B / C ) ) ) e. QQ -> ( C x. ( |_ ` ( B / C ) ) ) e. CC )
18	16, 17	syl 14	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. CC )
19	3, 6, 18	subdid 8312	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B - ( C x. ( |_ ` ( B / C ) ) ) ) ) = ( ( A x. B ) - ( A x. ( C x. ( |_ ` ( B / C ) ) ) ) ) )
20		qcn 9572	. . . . . . . . 9 |- ( C e. QQ -> C e. CC )
21	7, 20	syl 14	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. CC )
22		qre 9563	. . . . . . . . . 10 |- ( C e. QQ -> C e. RR )
23	7, 22	syl 14	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. RR )
24	23, 8	gt0ap0d 8527	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C # 0 )
25		qre 9563	. . . . . . . . . 10 |- ( A e. QQ -> A e. RR )
26	1, 25	syl 14	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. RR )
27		simp1r 1012	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < A )
28	26, 27	gt0ap0d 8527	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A # 0 )
29	6, 21, 3, 24, 28	divcanap5d 8713	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A x. B ) / ( A x. C ) ) = ( B / C ) )
30	29	fveq2d 5490	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) = ( |_ ` ( B / C ) ) )
31	30	oveq2d 5858	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) = ( ( A x. C ) x. ( |_ ` ( B / C ) ) ) )
32	12	zcnd 9314	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. CC )
33	3, 21, 32	mulassd 7922	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A x. C ) x. ( |_ ` ( B / C ) ) ) = ( A x. ( C x. ( |_ ` ( B / C ) ) ) ) )
34	31, 33	eqtr2d 2199	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( C x. ( |_ ` ( B / C ) ) ) ) = ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) )
35	34	oveq2d 5858	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A x. B ) - ( A x. ( C x. ( |_ ` ( B / C ) ) ) ) ) = ( ( A x. B ) - ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) ) )
36	19, 35	eqtrd 2198	. 2 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B - ( C x. ( |_ ` ( B / C ) ) ) ) ) = ( ( A x. B ) - ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) ) )
37		modqval 10259	. . . 4 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
38	4, 7, 8, 37	syl3anc 1228	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
39	38	oveq2d 5858	. 2 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B mod C ) ) = ( A x. ( B - ( C x. ( |_ ` ( B / C ) ) ) ) ) )
40		qmulcl 9575	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
41	1, 4, 40	syl2anc 409	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. B ) e. QQ )
42		qmulcl 9575	. . . 4 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
43	1, 7, 42	syl2anc 409	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. C ) e. QQ )
44	26, 23, 27, 8	mulgt0d 8021	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < ( A x. C ) )
45		modqval 10259	. . 3 |- ( ( ( A x. B ) e. QQ /\ ( A x. C ) e. QQ /\ 0 < ( A x. C ) ) -> ( ( A x. B ) mod ( A x. C ) ) = ( ( A x. B ) - ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) ) )
46	41, 43, 44, 45	syl3anc 1228	. 2 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A x. B ) mod ( A x. C ) ) = ( ( A x. B ) - ( ( A x. C ) x. ( |_ ` ( ( A x. B ) / ( A x. C ) ) ) ) ) )
47	36, 39, 46	3eqtr4d 2208	1 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   < clt 7933   - cmin 8069   / cdiv 8568   ZZcz 9191   QQcq 9557   |_cfl 10203   mod cmo 10257

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258

This theorem is referenced by: (None)