Theorem pnncan 8513

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
pnncan	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( B + C ) )

Proof of Theorem pnncan

Step	Hyp	Ref	Expression
1		simp1 1024	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp2 1025	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
3	1, 2	addcld 8292	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
4		simp3 1026	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		subsub 8502	. . 3 |- ( ( ( A + B ) e. CC /\ A e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( ( ( A + B ) - A ) + C ) )
6	3, 1, 4, 5	syl3anc 1274	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( ( ( A + B ) - A ) + C ) )
7		pncan2 8479	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
8	7	3adant3 1044	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - A ) = B )
9	8	oveq1d 6064	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) + C ) = ( B + C ) )
10	6, 9	eqtrd 2265	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( B + C ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203  (class class class)co 6049   CCcc 8124   + caddc 8129   - cmin 8443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-setind 4658  ax-resscn 8218  ax-1cn 8219  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-sub 8445

This theorem is referenced by:  ppncan  8514  pnncani  8567  pnncand  8622  halfaddsub  9471  shftval2  11507  sinmul  12426  pythagtriplem4  12962  pythagtriplem16  12973