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Theorem congmul 27023
Description: If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Assertion
Ref Expression
congmul  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  x.  D )  -  ( C  x.  E )
) )

Proof of Theorem congmul
StepHypRef Expression
1 simp11 987 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  e.  ZZ )
2 simp12 988 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  B  e.  ZZ )
3 simp2l 983 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  D  e.  ZZ )
42, 3zmulcld 10373 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( B  x.  D )  e.  ZZ )
5 simp2r 984 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  E  e.  ZZ )
62, 5zmulcld 10373 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( B  x.  E )  e.  ZZ )
7 simp13 989 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  C  e.  ZZ )
87, 5zmulcld 10373 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( C  x.  E )  e.  ZZ )
9 simp3r 986 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( D  -  E
) )
10 zsubcl 10311 . . . . . 6  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  ( D  -  E
)  e.  ZZ )
11103ad2ant2 979 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( D  -  E )  e.  ZZ )
12 dvdsmultr2 12877 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( D  -  E )  e.  ZZ )  ->  ( A  ||  ( D  -  E )  ->  A  ||  ( B  x.  ( D  -  E )
) ) )
131, 2, 11, 12syl3anc 1184 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( A  ||  ( D  -  E )  ->  A  ||  ( B  x.  ( D  -  E )
) ) )
149, 13mpd 15 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( B  x.  ( D  -  E )
) )
15 zcn 10279 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
16153ad2ant2 979 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
17163ad2ant1 978 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  B  e.  CC )
18 zcn 10279 . . . . . 6  |-  ( D  e.  ZZ  ->  D  e.  CC )
1918adantr 452 . . . . 5  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  D  e.  CC )
20193ad2ant2 979 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  D  e.  CC )
21 zcn 10279 . . . . . 6  |-  ( E  e.  ZZ  ->  E  e.  CC )
2221adantl 453 . . . . 5  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  E  e.  CC )
23223ad2ant2 979 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  E  e.  CC )
2417, 20, 23subdid 9481 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( B  x.  ( D  -  E ) )  =  ( ( B  x.  D )  -  ( B  x.  E )
) )
2514, 24breqtrd 4228 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  x.  D )  -  ( B  x.  E )
) )
26 simp3l 985 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( B  -  C
) )
272, 7zsubcld 10372 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( B  -  C )  e.  ZZ )
28 dvdsmultr1 12876 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  E  e.  ZZ )  ->  ( A  ||  ( B  -  C )  ->  A  ||  ( ( B  -  C )  x.  E ) ) )
291, 27, 5, 28syl3anc 1184 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( A  ||  ( B  -  C )  ->  A  ||  ( ( B  -  C )  x.  E
) ) )
3026, 29mpd 15 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  -  C )  x.  E
) )
31 zcn 10279 . . . . . 6  |-  ( C  e.  ZZ  ->  C  e.  CC )
32313ad2ant3 980 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
33323ad2ant1 978 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  C  e.  CC )
3417, 33, 23subdird 9482 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  (
( B  -  C
)  x.  E )  =  ( ( B  x.  E )  -  ( C  x.  E
) ) )
3530, 34breqtrd 4228 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  x.  E )  -  ( C  x.  E )
) )
36 congtr 27021 . 2  |-  ( ( ( A  e.  ZZ  /\  ( B  x.  D
)  e.  ZZ )  /\  ( ( B  x.  E )  e.  ZZ  /\  ( C  x.  E )  e.  ZZ )  /\  ( A  ||  ( ( B  x.  D )  -  ( B  x.  E
) )  /\  A  ||  ( ( B  x.  E )  -  ( C  x.  E )
) ) )  ->  A  ||  ( ( B  x.  D )  -  ( C  x.  E
) ) )
371, 4, 6, 8, 25, 35, 36syl222anc 1200 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  x.  D )  -  ( C  x.  E )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8980    x. cmul 8987    - cmin 9283   ZZcz 10274    || cdivides 12844
This theorem is referenced by:  mzpcong  27028  jm2.18  27050  jm2.15nn0  27065  jm2.16nn0  27066  jm2.27c  27069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-dvds 12845
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