Theorem apadd1 7599

Description: Addition respects apartness. Analogue of addcan 7191 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)

Assertion

Ref	Expression
apadd1	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )

Proof of Theorem apadd1

Dummy variables u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 7023	. . 3 |- ( C e. CC -> E. u e. RR E. v e. RR C = ( u + ( _i x. v ) ) )
2	1	3ad2ant3 927	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. u e. RR E. v e. RR C = ( u + ( _i x. v ) ) )
3		cnre 7023	. . . . . . 7 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
4	3	3ad2ant2 926	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
5	4	ad2antrr 457	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
6		cnre 7023	. . . . . . . . . . 11 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
7	6	3ad2ant1 925	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
8	7	ad2antrr 457	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
9	8	ad2antrr 457	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
10		simplrl 487	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. RR )
11		simplrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. RR )
12		simprl 483	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> z e. RR )
13	12	ad3antrrr 461	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. RR )
14		simprr 484	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> w e. RR )
15	14	ad3antrrr 461	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. RR )
16		apreim 7594	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
17	10, 11, 13, 15, 16	syl22anc 1136	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
18		simpr 103	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> A = ( x + ( _i x. y ) ) )
19		simpllr 486	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> B = ( z + ( _i x. w ) ) )
20	18, 19	breq12d 3777	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
21		simprl 483	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) -> u e. RR )
22	21	ad2antrr 457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> u e. RR )
23	22	ad3antrrr 461	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> u e. RR )
24	10, 23	readdcld 7055	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x + u ) e. RR )
25		simprr 484	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) -> v e. RR )
26	25	ad2antrr 457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> v e. RR )
27	26	ad3antrrr 461	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> v e. RR )
28	11, 27	readdcld 7055	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y + v ) e. RR )
29	13, 23	readdcld 7055	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z + u ) e. RR )
30	15, 27	readdcld 7055	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w + v ) e. RR )
31		apreim 7594	. . . . . . . . . . . . 13 |- ( ( ( ( x + u ) e. RR /\ ( y + v ) e. RR ) /\ ( ( z + u ) e. RR /\ ( w + v ) e. RR ) ) -> ( ( ( x + u ) + ( _i x. ( y + v ) ) ) # ( ( z + u ) + ( _i x. ( w + v ) ) ) <-> ( ( x + u ) # ( z + u ) \/ ( y + v ) # ( w + v ) ) ) )
32	24, 28, 29, 30, 31	syl22anc 1136	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( ( x + u ) + ( _i x. ( y + v ) ) ) # ( ( z + u ) + ( _i x. ( w + v ) ) ) <-> ( ( x + u ) # ( z + u ) \/ ( y + v ) # ( w + v ) ) ) )
33	10	recnd 7054	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. CC )
34		ax-icn 6979	. . . . . . . . . . . . . . . . 17 |- _i e. CC
35	34	a1i 9	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> _i e. CC )
36	11	recnd 7054	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. CC )
37	35, 36	mulcld 7047	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. y ) e. CC )
38	23	recnd 7054	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> u e. CC )
39	27	recnd 7054	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> v e. CC )
40	35, 39	mulcld 7047	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. v ) e. CC )
41	33, 37, 38, 40	add4d 7180	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) + ( u + ( _i x. v ) ) ) = ( ( x + u ) + ( ( _i x. y ) + ( _i x. v ) ) ) )
42		simplr 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> C = ( u + ( _i x. v ) ) )
43	42	ad3antrrr 461	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> C = ( u + ( _i x. v ) ) )
44	18, 43	oveq12d 5530	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A + C ) = ( ( x + ( _i x. y ) ) + ( u + ( _i x. v ) ) ) )
45	35, 36, 39	adddid 7051	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. ( y + v ) ) = ( ( _i x. y ) + ( _i x. v ) ) )
46	45	oveq2d 5528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + u ) + ( _i x. ( y + v ) ) ) = ( ( x + u ) + ( ( _i x. y ) + ( _i x. v ) ) ) )
47	41, 44, 46	3eqtr4d 2082	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A + C ) = ( ( x + u ) + ( _i x. ( y + v ) ) ) )
48	13	recnd 7054	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. CC )
49	15	recnd 7054	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. CC )
50	35, 49	mulcld 7047	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. w ) e. CC )
51	48, 50, 38, 40	add4d 7180	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( z + ( _i x. w ) ) + ( u + ( _i x. v ) ) ) = ( ( z + u ) + ( ( _i x. w ) + ( _i x. v ) ) ) )
52	19, 43	oveq12d 5530	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( B + C ) = ( ( z + ( _i x. w ) ) + ( u + ( _i x. v ) ) ) )
53	35, 49, 39	adddid 7051	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. ( w + v ) ) = ( ( _i x. w ) + ( _i x. v ) ) )
54	53	oveq2d 5528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( z + u ) + ( _i x. ( w + v ) ) ) = ( ( z + u ) + ( ( _i x. w ) + ( _i x. v ) ) ) )
55	51, 52, 54	3eqtr4d 2082	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( B + C ) = ( ( z + u ) + ( _i x. ( w + v ) ) ) )
56	47, 55	breq12d 3777	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( A + C ) # ( B + C ) <-> ( ( x + u ) + ( _i x. ( y + v ) ) ) # ( ( z + u ) + ( _i x. ( w + v ) ) ) ) )
57		reapadd1 7587	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ z e. RR /\ u e. RR ) -> ( x # z <-> ( x + u ) # ( z + u ) ) )
58	10, 13, 23, 57	syl3anc 1135	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> ( x + u ) # ( z + u ) ) )
59		reapadd1 7587	. . . . . . . . . . . . . 14 |- ( ( y e. RR /\ w e. RR /\ v e. RR ) -> ( y # w <-> ( y + v ) # ( w + v ) ) )
60	11, 15, 27, 59	syl3anc 1135	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> ( y + v ) # ( w + v ) ) )
61	58, 60	orbi12d 707	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> ( ( x + u ) # ( z + u ) \/ ( y + v ) # ( w + v ) ) ) )
62	32, 56, 61	3bitr4d 209	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( A + C ) # ( B + C ) <-> ( x # z \/ y # w ) ) )
63	17, 20, 62	3bitr4d 209	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
64	63	ex 108	. . . . . . . . 9 |- ( ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
65	64	rexlimdvva 2440	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
66	9, 65	mpd 13	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
67	66	ex 108	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
68	67	rexlimdvva 2440	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
69	5, 68	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) /\ C = ( u + ( _i x. v ) ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
70	69	ex 108	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( u e. RR /\ v e. RR ) ) -> ( C = ( u + ( _i x. v ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
71	70	rexlimdvva 2440	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( E. u e. RR E. v e. RR C = ( u + ( _i x. v ) ) -> ( A # B <-> ( A + C ) # ( B + C ) ) ) )
72	2, 71	mpd 13	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   E.wrex 2307   class class class wbr 3764  (class class class)co 5512   CCcc 6887   RRcr 6888   _ici 6891   + caddc 6892   x. cmul 6894   # cap 7572

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-ltxr 7065  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573

This theorem is referenced by:  apadd2  7600  addext  7601  subap0d  7631