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Theorem nnncan1 8111
Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
nnncan1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  ( A  -  C ) )  =  ( C  -  B ) )

Proof of Theorem nnncan1
StepHypRef Expression
1 subcl 8074 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
213adant2 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
3 sub32 8109 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  -  C )  e.  CC )  ->  (
( A  -  B
)  -  ( A  -  C ) )  =  ( ( A  -  ( A  -  C ) )  -  B ) )
42, 3syld3an3 1265 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  ( A  -  C ) )  =  ( ( A  -  ( A  -  C ) )  -  B ) )
5 nncan 8104 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( A  -  C )
)  =  C )
653adant2 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( A  -  C ) )  =  C )
76oveq1d 5839 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  ( A  -  C )
)  -  B )  =  ( C  -  B ) )
84, 7eqtrd 2190 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  ( A  -  C ) )  =  ( C  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    = wceq 1335    e. wcel 2128  (class class class)co 5824   CCcc 7730    - cmin 8046
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-sub 8048
This theorem is referenced by:  nnncan1d  8220  cju  8832
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