Theorem addsubeq4 7390

Description: Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Assertion

Ref	Expression
addsubeq4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )

Proof of Theorem addsubeq4

Step	Hyp	Ref	Expression
1		eqcom 2084	. . 3 |- ( ( C - A ) = ( B - D ) <-> ( B - D ) = ( C - A ) )
2		subcl 7374	. . . . . 6 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 264	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4		subadd 7378	. . . . . . 7 |- ( ( B e. CC /\ D e. CC /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
5	4	3expa 1139	. . . . . 6 |- ( ( ( B e. CC /\ D e. CC ) /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
6	5	ancoms 264	. . . . 5 |- ( ( ( C - A ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
7	3, 6	sylan 277	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
8	7	an4s 553	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
9	1, 8	syl5bb 190	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C - A ) = ( B - D ) <-> ( D + ( C - A ) ) = B ) )
10		addcom 7312	. . . . . . 7 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
11	10	adantl 271	. . . . . 6 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( C + D ) = ( D + C ) )
12	11	oveq1d 5558	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( ( D + C ) - A ) )
13		addsubass 7385	. . . . . . . 8 |- ( ( D e. CC /\ C e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
14	13	3com12 1143	. . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
15	14	3expa 1139	. . . . . 6 |- ( ( ( C e. CC /\ D e. CC ) /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
16	15	ancoms 264	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
17	12, 16	eqtrd 2114	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
18	17	adantlr 461	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
19	18	eqeq1d 2090	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( D + ( C - A ) ) = B ) )
20		addcl 7160	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
21		subadd 7378	. . . . 5 |- ( ( ( C + D ) e. CC /\ A e. CC /\ B e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
22	21	3expb 1140	. . . 4 |- ( ( ( C + D ) e. CC /\ ( A e. CC /\ B e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
23	22	ancoms 264	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
24	20, 23	sylan2 280	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
25	9, 19, 24	3bitr2rd 215	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434  (class class class)co 5543   CCcc 7041   + caddc 7046   - cmin 7346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348

This theorem is referenced by:  subcan  7430  addsubeq4d  7537