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Theorem congsub 26205
Description: If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Assertion
Ref Expression
congsub  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  -  D )  -  ( C  -  E )
) )

Proof of Theorem congsub
StepHypRef Expression
1 simp11 985 . . 3  |-  ( ( ( 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 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 )
4 simp2l 981 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  D  e.  ZZ )
54znegcld 10166 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  -u D  e.  ZZ )
6 simp2r 982 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  E  e.  ZZ )
76znegcld 10166 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  -u E  e.  ZZ )
8 simp3l 983 . . 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
) )
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 congneg 26204 . . . 4  |-  ( ( ( A  e.  ZZ  /\  D  e.  ZZ )  /\  ( E  e.  ZZ  /\  A  ||  ( D  -  E
) ) )  ->  A  ||  ( -u D  -  -u E ) )
111, 4, 6, 9, 10syl22anc 1183 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( -u D  -  -u E ) )
12 congadd 26201 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( -u D  e.  ZZ  /\  -u E  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( -u D  -  -u E ) ) )  ->  A  ||  (
( B  +  -u D )  -  ( C  +  -u E ) ) )
131, 2, 3, 5, 7, 8, 11, 12syl322anc 1210 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  +  -u D )  -  ( C  +  -u E ) ) )
142zcnd 10165 . . . 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 )
154zcnd 10165 . . . 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 )
1614, 15negsubd 9208 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( B  +  -u D )  =  ( B  -  D ) )
173zcnd 10165 . . . 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 )
186zcnd 10165 . . . 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 )
1917, 18negsubd 9208 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  ( C  +  -u E )  =  ( C  -  E ) )
2016, 19oveq12d 5918 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  (
( B  +  -u D )  -  ( C  +  -u E ) )  =  ( ( B  -  D )  -  ( C  -  E ) ) )
2113, 20breqtrd 4084 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( D  -  E )
) )  ->  A  ||  ( ( B  -  D )  -  ( C  -  E )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1701   class class class wbr 4060  (class class class)co 5900    + caddc 8785    - cmin 9082   -ucneg 9083   ZZcz 10071    || cdivides 12578
This theorem is referenced by:  jm2.18  26229  jm2.15nn0  26244  jm2.16nn0  26245  jm2.27c  26248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-dvds 12579
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