Theorem modqdi 10158

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 1005	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2		qcn 9419	. . . . 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 982	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
5		qcn 9419	. . . . 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 1009	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
8		simp3r 1010	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
9	8	gt0ne0d 8267	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C =/= 0 )
10		qdivcl 9428	. . . . . . . . 9 |- ( ( B e. QQ /\ C e. QQ /\ C =/= 0 ) -> ( B / C ) e. QQ )
11	4, 7, 9, 10	syl3anc 1216	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B / C ) e. QQ )
12	11	flqcld 10043	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. ZZ )
13		zq 9411	. . . . . . 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 9422	. . . . . 6 |- ( ( C e. QQ /\ ( |_ ` ( B / C ) ) e. QQ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
16	7, 14, 15	syl2anc 408	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
17		qcn 9419	. . . . 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 8169	. . 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 9419	. . . . . . . . 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 9410	. . . . . . . . . 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 8384	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C # 0 )
25		qre 9410	. . . . . . . . . 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 1006	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < A )
28	26, 27	gt0ap0d 8384	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A # 0 )
29	6, 21, 3, 24, 28	divcanap5d 8570	. . . . . . 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 5418	. . . . . 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 5783	. . . . 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 9167	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. CC )
33	3, 21, 32	mulassd 7782	. . . . 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 2171	. . . 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 5783	. . 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 2170	. 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 10090	. . . 4 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
38	4, 7, 8, 37	syl3anc 1216	. . 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 5783	. 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 9422	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
41	1, 4, 40	syl2anc 408	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. B ) e. QQ )
42		qmulcl 9422	. . . 4 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
43	1, 7, 42	syl2anc 408	. . 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 7878	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < ( A x. C ) )
45		modqval 10090	. . 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 1216	. 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 2180	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 962   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   x. cmul 7618   < clt 7793   - cmin 7926   / cdiv 8425   ZZcz 9047   QQcq 9404   |_cfl 10034   mod cmo 10088

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-q 9405  df-rp 9435  df-fl 10036  df-mod 10089

This theorem is referenced by: (None)