Theorem addsubeq4 8189

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 2190	. . 3 |- ( ( C - A ) = ( B - D ) <-> ( B - D ) = ( C - A ) )
2		subcl 8173	. . . . . 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 8177	. . . . . . 7 |- ( ( B e. CC /\ D e. CC /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
5	4	3expa 1204	. . . . . 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 8111	. . . . . . 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 5905	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( ( D + C ) - A ) )
13		addsubass 8184	. . . . . . . 8 |- ( ( D e. CC /\ C e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
14	13	3com12 1208	. . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
15	14	3expa 1204	. . . . . 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 2221	. . . 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 2197	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( D + ( C - A ) ) = B ) )
20		addcl 7953	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
21		subadd 8177	. . . . 5 |- ( ( ( C + D ) e. CC /\ A e. CC /\ B e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
22	21	3expb 1205	. . . 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 1363   e. wcel 2159  (class class class)co 5890   CCcc 7826   + caddc 7831   - cmin 8145

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-setind 4550  ax-resscn 7920  ax-1cn 7921  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-sub 8147

This theorem is referenced by:  subcan  8229  addsubeq4d  8336