Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  congmul Unicode version

Theorem congmul 26716
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 10306 . 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 10306 . 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 10306 . 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 10244 . . . . . 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 12805 . . . . 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 10212 . . . . . 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 10212 . . . . . 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 10212 . . . . . 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 9414 . . 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 4170 . 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 10305 . . . . 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 12804 . . . . 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 10212 . . . . . 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 9415 . . 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 4170 . 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 26714 . 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 1717   class class class wbr 4146  (class class class)co 6013   CCcc 8914    x. cmul 8921    - cmin 9216   ZZcz 10207    || cdivides 12772
This theorem is referenced by:  mzpcong  26721  jm2.18  26743  jm2.15nn0  26758  jm2.16nn0  26759  jm2.27c  26762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-dvds 12773
  Copyright terms: Public domain W3C validator