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Theorem subsub4 8002
Description: Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
subsub4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  C )  =  ( A  -  ( B  +  C
) ) )

Proof of Theorem subsub4
StepHypRef Expression
1 nppcan2 8000 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  ( B  +  C )
)  +  C )  =  ( A  -  B ) )
2 simp1 981 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
3 simp2 982 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
4 subcl 7968 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
52, 3, 4syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
6 simp3 983 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
73, 6addcld 7792 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
8 subcl 7968 . . . 4  |-  ( ( A  e.  CC  /\  ( B  +  C
)  e.  CC )  ->  ( A  -  ( B  +  C
) )  e.  CC )
92, 7, 8syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  +  C ) )  e.  CC )
10 subadd2 7973 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  C  e.  CC  /\  ( A  -  ( B  +  C ) )  e.  CC )  ->  (
( ( A  -  B )  -  C
)  =  ( A  -  ( B  +  C ) )  <->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) ) )
115, 6, 9, 10syl3anc 1216 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  =  ( A  -  ( B  +  C ) )  <->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) ) )
121, 11mpbird 166 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  C )  =  ( A  -  ( B  +  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7625    + caddc 7630    - cmin 7940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7942
This theorem is referenced by:  sub32  8003  nnncan  8004  pnpcan  8008  addsub4  8012  subsub4d  8111  2shfti  10610  nn0seqcvgd  11728
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