Theorem 2addsub 8145

Description: Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)

Assertion

Ref	Expression
2addsub	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) )

Proof of Theorem 2addsub

Step	Hyp	Ref	Expression
1		add32 8090	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
2	1	3expa 1203	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
3	2	adantrr 479	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
4	3	oveq1d 5880	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) + B ) - D ) )
5		addcl 7911	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + C ) e. CC )
6		addsub 8142	. . . . 5 |- ( ( ( A + C ) e. CC /\ B e. CC /\ D e. CC ) -> ( ( ( A + C ) + B ) - D ) = ( ( ( A + C ) - D ) + B ) )
7	6	3expb 1204	. . . 4 |- ( ( ( A + C ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( ( A + C ) + B ) - D ) = ( ( ( A + C ) - D ) + B ) )
8	5, 7	sylan 283	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( ( A + C ) + B ) - D ) = ( ( ( A + C ) - D ) + B ) )
9	8	an4s 588	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + C ) + B ) - D ) = ( ( ( A + C ) - D ) + B ) )
10	4, 9	eqtrd 2208	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146  (class class class)co 5865   CCcc 7784   + caddc 7789   - cmin 8102

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530  ax-resscn 7878  ax-1cn 7879  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-sub 8104

This theorem is referenced by:  2addsubd  8292