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Theorem subsub4 9843
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 9841 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  ( B  +  C )
)  +  C )  =  ( A  -  B ) )
2 simp1 991 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
3 simp2 992 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
4 subcl 9810 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
52, 3, 4syl2anc 661 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
6 simp3 993 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
73, 6addcld 9606 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
8 subcl 9810 . . . 4  |-  ( ( A  e.  CC  /\  ( B  +  C
)  e.  CC )  ->  ( A  -  ( B  +  C
) )  e.  CC )
92, 7, 8syl2anc 661 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  +  C ) )  e.  CC )
10 subadd2 9815 . . 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 1223 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  =  ( A  -  ( B  +  C ) )  <->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) ) )
121, 11mpbird 232 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  C )  =  ( A  -  ( B  +  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762  (class class class)co 6277   CCcc 9481    + caddc 9486    - cmin 9796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-ltxr 9624  df-sub 9798
This theorem is referenced by:  sub32  9844  nnncan  9845  pnpcan  9849  addsub4  9853  subsub4d  9952  2shfti  12865  divalglem2  13903  nn0seqcvgd  14049  plydivlem4  22421  ax5seglem7  23909  itg2addnclem3  29634
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