Theorem modqmul12d 10744

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 9961	. . . 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 9961	. . . 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 10743	. 2 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. C ) mod E ) )
12	4	zcnd 9704	. . . . 5 |- ( ph -> B e. CC )
13	7	zcnd 9704	. . . . 5 |- ( ph -> C e. CC )
14	12, 13	mulcomd 8297	. . . 4 |- ( ph -> ( B x. C ) = ( C x. B ) )
15	14	oveq1d 6067	. . 3 |- ( ph -> ( ( B x. C ) mod E ) = ( ( C x. B ) mod E ) )
16		zq 9961	. . . . 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 9961	. . . . 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 10743	. . 3 |- ( ph -> ( ( C x. B ) mod E ) = ( ( D x. B ) mod E ) )
23	18	zcnd 9704	. . . . 5 |- ( ph -> D e. CC )
24	23, 12	mulcomd 8297	. . . 4 |- ( ph -> ( D x. B ) = ( B x. D ) )
25	24	oveq1d 6067	. . 3 |- ( ph -> ( ( D x. B ) mod E ) = ( ( B x. D ) mod E ) )
26	15, 22, 25	3eqtrd 2271	. 2 |- ( ph -> ( ( B x. C ) mod E ) = ( ( B x. D ) mod E ) )
27	11, 26	eqtrd 2267	1 |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   0cc0 8129   x. cmul 8134   < clt 8310   ZZcz 9579   QQcq 9954   mod cmo 10688

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-n0 9499  df-z 9580  df-q 9955  df-rp 9990  df-fl 10634  df-mod 10689

This theorem is referenced by:  modqexp  11032  fprodmodd  12331  modxai  13118  lgsdir2lem5  15922  lgseisenlem2  15961  lgseisenlem3  15962