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Theorem subsub2 7958
Description: Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
subsub2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )

Proof of Theorem subsub2
StepHypRef Expression
1 subcl 7929 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
213adant1 984 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
3 simp1 966 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
4 simp3 968 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
5 simp2 967 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
6 subcl 7929 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
74, 5, 6syl2anc 408 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  -  B )  e.  CC )
82, 3, 7add12d 7897 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  ( A  +  ( C  -  B ) ) )  =  ( A  +  ( ( B  -  C )  +  ( C  -  B ) ) ) )
9 npncan2 7957 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  ( C  -  B ) )  =  0 )
1093adant1 984 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  ( C  -  B ) )  =  0 )
1110oveq2d 5758 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( ( B  -  C )  +  ( C  -  B ) ) )  =  ( A  + 
0 ) )
123addid1d 7879 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  0 )  =  A )
138, 11, 123eqtrd 2154 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  ( A  +  ( C  -  B ) ) )  =  A )
143, 7addcld 7753 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( C  -  B ) )  e.  CC )
15 subadd 7933 . . 3  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC  /\  ( A  +  ( C  -  B )
)  e.  CC )  ->  ( ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) )  <->  ( ( B  -  C )  +  ( A  +  ( C  -  B ) ) )  =  A ) )
163, 2, 14, 15syl3anc 1201 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  ( B  -  C )
)  =  ( A  +  ( C  -  B ) )  <->  ( ( B  -  C )  +  ( A  +  ( C  -  B
) ) )  =  A ) )
1713, 16mpbird 166 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586   0cc0 7588    + caddc 7591    - cmin 7901
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903
This theorem is referenced by:  nncan  7959  subsub  7960  subsub3  7962  ppncan  7972  subadd4  7974  subsub2d  8070
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