Theorem modqdi 10058

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 988	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2		qcn 9328	. . . . 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 965	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
5		qcn 9328	. . . . 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 992	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
8		simp3r 993	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
9	8	gt0ne0d 8193	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C =/= 0 )
10		qdivcl 9337	. . . . . . . . 9 |- ( ( B e. QQ /\ C e. QQ /\ C =/= 0 ) -> ( B / C ) e. QQ )
11	4, 7, 9, 10	syl3anc 1199	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B / C ) e. QQ )
12	11	flqcld 9943	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. ZZ )
13		zq 9320	. . . . . . 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 9331	. . . . . 6 |- ( ( C e. QQ /\ ( |_ ` ( B / C ) ) e. QQ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
16	7, 14, 15	syl2anc 406	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
17		qcn 9328	. . . . 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 8095	. . 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 9328	. . . . . . . . 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 9319	. . . . . . . . . 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 8309	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C # 0 )
25		qre 9319	. . . . . . . . . 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 989	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < A )
28	26, 27	gt0ap0d 8309	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A # 0 )
29	6, 21, 3, 24, 28	divcanap5d 8490	. . . . . . 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 5379	. . . . . 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 5744	. . . . 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 9078	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. CC )
33	3, 21, 32	mulassd 7713	. . . . 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 2148	. . . 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 5744	. . 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 2147	. 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 9990	. . . 4 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
38	4, 7, 8, 37	syl3anc 1199	. . 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 5744	. 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 9331	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
41	1, 4, 40	syl2anc 406	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. B ) e. QQ )
42		qmulcl 9331	. . . 4 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
43	1, 7, 42	syl2anc 406	. . 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 7808	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < ( A x. C ) )
45		modqval 9990	. . 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 1199	. 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 2157	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 945   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   x. cmul 7552   < clt 7724   - cmin 7856   / cdiv 8345   ZZcz 8958   QQcq 9313   |_cfl 9934   mod cmo 9988

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-fl 9936  df-mod 9989

This theorem is referenced by: (None)