Theorem modqdi 10850

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 1052	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2		qcn 10050	. . . . 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 1029	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
5		qcn 10050	. . . . 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 1056	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
8		simp3r 1057	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
9	8	gt0ne0d 8847	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C =/= 0 )
10		qdivcl 10059	. . . . . . . . 9 |- ( ( B e. QQ /\ C e. QQ /\ C =/= 0 ) -> ( B / C ) e. QQ )
11	4, 7, 9, 10	syl3anc 1278	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( B / C ) e. QQ )
12	11	flqcld 10731	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. ZZ )
13		zq 10042	. . . . . . 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 10053	. . . . . 6 |- ( ( C e. QQ /\ ( |_ ` ( B / C ) ) e. QQ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
16	7, 14, 15	syl2anc 415	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( C x. ( |_ ` ( B / C ) ) ) e. QQ )
17		qcn 10050	. . . . 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 8748	. . 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 10050	. . . . . . . . 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 10041	. . . . . . . . . 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 8965	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> C # 0 )
25		qre 10041	. . . . . . . . . 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 1053	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < A )
28	26, 27	gt0ap0d 8965	. . . . . . . 8 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> A # 0 )
29	6, 21, 3, 24, 28	divcanap5d 9155	. . . . . . 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 5702	. . . . . 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 6104	. . . . 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 9780	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( |_ ` ( B / C ) ) e. CC )
33	3, 21, 32	mulassd 8355	. . . . 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 2272	. . . 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 6104	. . 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 2271	. 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 10782	. . . 4 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
38	4, 7, 8, 37	syl3anc 1278	. . 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 6104	. 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 10053	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
41	1, 4, 40	syl2anc 415	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. B ) e. QQ )
42		qmulcl 10053	. . . 4 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
43	1, 7, 42	syl2anc 415	. . 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 8456	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> 0 < ( A x. C ) )
45		modqval 10782	. . 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 1278	. 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 2281	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   x. cmul 8190   < clt 8366   - cmin 8504   / cdiv 9010   ZZcz 9655   QQcq 10035   |_cfl 10720   mod cmo 10780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781

This theorem is used by: (None)