Theorem modqmul12d 10612

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 9833	. . . 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 9833	. . . 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 10611	. 2 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. C ) mod E ) )
12	4	zcnd 9581	. . . . 5 |- ( ph -> B e. CC )
13	7	zcnd 9581	. . . . 5 |- ( ph -> C e. CC )
14	12, 13	mulcomd 8179	. . . 4 |- ( ph -> ( B x. C ) = ( C x. B ) )
15	14	oveq1d 6022	. . 3 |- ( ph -> ( ( B x. C ) mod E ) = ( ( C x. B ) mod E ) )
16		zq 9833	. . . . 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 9833	. . . . 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 10611	. . 3 |- ( ph -> ( ( C x. B ) mod E ) = ( ( D x. B ) mod E ) )
23	18	zcnd 9581	. . . . 5 |- ( ph -> D e. CC )
24	23, 12	mulcomd 8179	. . . 4 |- ( ph -> ( D x. B ) = ( B x. D ) )
25	24	oveq1d 6022	. . 3 |- ( ph -> ( ( D x. B ) mod E ) = ( ( B x. D ) mod E ) )
26	15, 22, 25	3eqtrd 2266	. 2 |- ( ph -> ( ( B x. C ) mod E ) = ( ( B x. D ) mod E ) )
27	11, 26	eqtrd 2262	1 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   0cc0 8010   x. cmul 8015   < clt 8192   ZZcz 9457   QQcq 9826   mod cmo 10556

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-q 9827  df-rp 9862  df-fl 10502  df-mod 10557

This theorem is referenced by:  modqexp  10900  fprodmodd  12167  modxai  12954  lgsdir2lem5  15726  lgseisenlem2  15765  lgseisenlem3  15766