Theorem modqmul12d 9380

Description: Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)

Hypotheses

Ref	Expression
modqmul12d.1	|- ( ph -> A e. ZZ )
modqmul12d.2	|- ( ph -> B e. ZZ )
modqmul12d.3	|- ( ph -> C e. ZZ )
modqmul12d.4	|- ( ph -> D e. ZZ )
modqmul12d.5	|- ( ph -> E e. QQ )
modqmul12d.egt0	|- ( ph -> 0 < E )
modqmul12d.6	|- ( ph -> ( A mod E ) = ( B mod E ) )
modqmul12d.7	|- ( ph -> ( C mod E ) = ( D mod E ) )

Assertion

Ref	Expression
modqmul12d	|- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )

Proof of Theorem modqmul12d

Step	Hyp	Ref	Expression
1		modqmul12d.1	. . . 4 |- ( ph -> A e. ZZ )
2		zq 8711	. . . 4 |- ( A e. ZZ -> A e. QQ )
3	1, 2	syl 14	. . 3 |- ( ph -> A e. QQ )
4		modqmul12d.2	. . . 4 |- ( ph -> B e. ZZ )
5		zq 8711	. . . 4 |- ( B e. ZZ -> B e. QQ )
6	4, 5	syl 14	. . 3 |- ( ph -> B e. QQ )
7		modqmul12d.3	. . 3 |- ( ph -> C e. ZZ )
8		modqmul12d.5	. . 3 |- ( ph -> E e. QQ )
9		modqmul12d.egt0	. . 3 |- ( ph -> 0 < E )
10		modqmul12d.6	. . 3 |- ( ph -> ( A mod E ) = ( B mod E ) )
11	3, 6, 7, 8, 9, 10	modqmul1 9379	. 2 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. C ) mod E ) )
12	4	zcnd 8470	. . . . 5 |- ( ph -> B e. CC )
13	7	zcnd 8470	. . . . 5 |- ( ph -> C e. CC )
14	12, 13	mulcomd 7140	. . . 4 |- ( ph -> ( B x. C ) = ( C x. B ) )
15	14	oveq1d 5547	. . 3 |- ( ph -> ( ( B x. C ) mod E ) = ( ( C x. B ) mod E ) )
16		zq 8711	. . . . 5 |- ( C e. ZZ -> C e. QQ )
17	7, 16	syl 14	. . . 4 |- ( ph -> C e. QQ )
18		modqmul12d.4	. . . . 5 |- ( ph -> D e. ZZ )
19		zq 8711	. . . . 5 |- ( D e. ZZ -> D e. QQ )
20	18, 19	syl 14	. . . 4 |- ( ph -> D e. QQ )
21		modqmul12d.7	. . . 4 |- ( ph -> ( C mod E ) = ( D mod E ) )
22	17, 20, 4, 8, 9, 21	modqmul1 9379	. . 3 |- ( ph -> ( ( C x. B ) mod E ) = ( ( D x. B ) mod E ) )
23	18	zcnd 8470	. . . . 5 |- ( ph -> D e. CC )
24	23, 12	mulcomd 7140	. . . 4 |- ( ph -> ( D x. B ) = ( B x. D ) )
25	24	oveq1d 5547	. . 3 |- ( ph -> ( ( D x. B ) mod E ) = ( ( B x. D ) mod E ) )
26	15, 22, 25	3eqtrd 2117	. 2 |- ( ph -> ( ( B x. C ) mod E ) = ( ( B x. D ) mod E ) )
27	11, 26	eqtrd 2113	1 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   0cc0 6981   x. cmul 6986   < clt 7153   ZZcz 8351   QQcq 8704   mod cmo 9324

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-q 8705  df-rp 8735  df-fl 9274  df-mod 9325

This theorem is referenced by: (None)