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Theorem subsub4 9907
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 9905 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  ( B  +  C )
)  +  C )  =  ( A  -  B ) )
2 simp1 1008 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
3 simp2 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
4 subcl 9874 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
52, 3, 4syl2anc 667 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
6 simp3 1010 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
73, 6addcld 9662 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
8 subcl 9874 . . . 4  |-  ( ( A  e.  CC  /\  ( B  +  C
)  e.  CC )  ->  ( A  -  ( B  +  C
) )  e.  CC )
92, 7, 8syl2anc 667 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  +  C ) )  e.  CC )
10 subadd2 9879 . . 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 1268 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  =  ( A  -  ( B  +  C ) )  <->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) ) )
121, 11mpbird 236 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 188    /\ w3a 985    = wceq 1444    e. wcel 1887  (class class class)co 6290   CCcc 9537    + caddc 9542    - cmin 9860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-ltxr 9680  df-sub 9862
This theorem is referenced by:  sub32  9908  nnncan  9909  pnpcan  9913  addsub4  9917  subsub4d  10017  2shfti  13143  divalglem2  14373  divalglem2OLD  14374  nn0seqcvgd  14529  plydivlem4  23249  ax5seglem7  24965  itg2addnclem3  31995
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