Theorem modqdi 10644

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 1045	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2		qcn 9858	. . . . 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 1022	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
5		qcn 9858	. . . . 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 1049	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
8		simp3r 1050	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
9	8	gt0ne0d 8682	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C =/= 0 )
10		qdivcl 9867	. . . . . . . . 9 |- ( ( B e. QQ /\ C e. QQ /\ C =/= 0 ) -> ( B / C ) e. QQ )
11	4, 7, 9, 10	syl3anc 1271	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B / C ) e. QQ )
12	11	flqcld 10527	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. ZZ )
13		zq 9850	. . . . . . 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 9861	. . . . . 6 |- ( ( C e. QQ /\ ( |_ ` ( B / C ) ) e. QQ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
16	7, 14, 15	syl2anc 411	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
17		qcn 9858	. . . . 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 8583	. . 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 9858	. . . . . . . . 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 9849	. . . . . . . . . 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 8799	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C # 0 )
25		qre 9849	. . . . . . . . . 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 1046	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < A )
28	26, 27	gt0ap0d 8799	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A # 0 )
29	6, 21, 3, 24, 28	divcanap5d 8987	. . . . . . 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 5639	. . . . . 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 6029	. . . . 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 9593	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. CC )
33	3, 21, 32	mulassd 8193	. . . . 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 2263	. . . 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 6029	. . 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 2262	. 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 10576	. . . 4 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
38	4, 7, 8, 37	syl3anc 1271	. . 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 6029	. 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 9861	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
41	1, 4, 40	syl2anc 411	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. B ) e. QQ )
42		qmulcl 9861	. . . 4 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
43	1, 7, 42	syl2anc 411	. . 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 8292	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < ( A x. C ) )
45		modqval 10576	. . 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 1271	. 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 2272	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   x. cmul 8027   < clt 8204   - cmin 8340   / cdiv 8842   ZZcz 9469   QQcq 9843   |_cfl 10518   mod cmo 10574

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-n0 9393  df-z 9470  df-q 9844  df-rp 9879  df-fl 10520  df-mod 10575

This theorem is referenced by: (None)