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Theorem congmul 27157
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 985 . 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 986 . . 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 981 . . 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 10139 . 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 982 . . 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 10139 . 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 987 . . 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 10139 . 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 984 . . . 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 10077 . . . . . 6  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  ( D  -  E
)  e.  ZZ )
11103ad2ant2 977 . . . . 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 12580 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( D  -  E )  e.  ZZ )  ->  ( A  ||  ( D  -  E )  ->  A  ||  ( B  x.  ( D  -  E )
) ) )
131, 2, 11, 12syl3anc 1182 . . . 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 14 . . 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 10045 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
16153ad2ant2 977 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
17163ad2ant1 976 . . . 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 10045 . . . . . 6  |-  ( D  e.  ZZ  ->  D  e.  CC )
1918adantr 451 . . . . 5  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  D  e.  CC )
20193ad2ant2 977 . . . 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 10045 . . . . . 6  |-  ( E  e.  ZZ  ->  E  e.  CC )
2221adantl 452 . . . . 5  |-  ( ( D  e.  ZZ  /\  E  e.  ZZ )  ->  E  e.  CC )
23223ad2ant2 977 . . . 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 9251 . . 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 4063 . 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 983 . . . 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 10138 . . . . 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 12579 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  E  e.  ZZ )  ->  ( A  ||  ( B  -  C )  ->  A  ||  ( ( B  -  C )  x.  E ) ) )
291, 27, 5, 28syl3anc 1182 . . . 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 14 . . 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 10045 . . . . . 6  |-  ( C  e.  ZZ  ->  C  e.  CC )
32313ad2ant3 978 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
33323ad2ant1 976 . . . 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 9252 . . 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 4063 . 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 27155 . 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 1198 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 358    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751    x. cmul 8758    - cmin 9053   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  mzpcong  27162  jm2.18  27184  jm2.15nn0  27199  jm2.16nn0  27200  jm2.27c  27203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-dvds 12548
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