Theorem addsubeq4 8174

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 2179	. . 3 |- ( ( C - A ) = ( B - D ) <-> ( B - D ) = ( C - A ) )
2		subcl 8158	. . . . . 6 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 268	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4		subadd 8162	. . . . . . 7 |- ( ( B e. CC /\ D e. CC /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
5	4	3expa 1203	. . . . . 6 |- ( ( ( B e. CC /\ D e. CC ) /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
6	5	ancoms 268	. . . . 5 |- ( ( ( C - A ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
7	3, 6	sylan 283	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
8	7	an4s 588	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
9	1, 8	bitrid 192	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C - A ) = ( B - D ) <-> ( D + ( C - A ) ) = B ) )
10		addcom 8096	. . . . . . 7 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
11	10	adantl 277	. . . . . 6 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( C + D ) = ( D + C ) )
12	11	oveq1d 5892	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( ( D + C ) - A ) )
13		addsubass 8169	. . . . . . . 8 |- ( ( D e. CC /\ C e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
14	13	3com12 1207	. . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
15	14	3expa 1203	. . . . . 6 |- ( ( ( C e. CC /\ D e. CC ) /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
16	15	ancoms 268	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
17	12, 16	eqtrd 2210	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
18	17	adantlr 477	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
19	18	eqeq1d 2186	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( D + ( C - A ) ) = B ) )
20		addcl 7938	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
21		subadd 8162	. . . . 5 |- ( ( ( C + D ) e. CC /\ A e. CC /\ B e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
22	21	3expb 1204	. . . 4 |- ( ( ( C + D ) e. CC /\ ( A e. CC /\ B e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
23	22	ancoms 268	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
24	20, 23	sylan2 286	. 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 217	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 104   <-> wb 105   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   + caddc 7816   - cmin 8130

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132

This theorem is referenced by:  subcan  8214  addsubeq4d  8321