Theorem addsrmo 9515

Description: There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
addsrmo	|- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )

Distinct variable groups:   t, A, u, v, w, z   t, B, u, v, w, z

Proof of Theorem addsrmo

Dummy variables f g h q s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 9507	. . . . . . . . . . . . . . . 16 |- ~R Er ( P. X. P. )
2	1	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ~R Er ( P. X. P. ) )
3		prsrlem1 9514	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) )
4		addcmpblnr 9511	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) -> ( ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) -> <. ( w +P. u ) , ( v +P. t ) >. ~R <. ( s +P. g ) , ( f +P. h ) >. ) )
5	4	imp 436	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) -> <. ( w +P. u ) , ( v +P. t ) >. ~R <. ( s +P. g ) , ( f +P. h ) >. )
6	3, 5	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. ( w +P. u ) , ( v +P. t ) >. ~R <. ( s +P. g ) , ( f +P. h ) >. )
7	2, 6	erthi 7428	. . . . . . . . . . . . . 14 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
8	7	adantrlr 737	. . . . . . . . . . . . 13 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
9	8	adantrrr 739	. . . . . . . . . . . 12 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
10		simprlr 781	. . . . . . . . . . . 12 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) ) -> z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R )
11		simprrr 783	. . . . . . . . . . . 12 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) ) -> q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
12	9, 10, 11	3eqtr4d 2515	. . . . . . . . . . 11 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) ) -> z = q )
13	12	expr 626	. . . . . . . . . 10 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> ( ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) )
14	13	exlimdvv 1788	. . . . . . . . 9 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> ( E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) )
15	14	exlimdvv 1788	. . . . . . . 8 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) )
16	15	ex 441	. . . . . . 7 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) ) )
17	16	exlimdvv 1788	. . . . . 6 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> ( E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) ) )
18	17	exlimdvv 1788	. . . . 5 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) -> z = q ) ) )
19	18	impd 438	. . . 4 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) -> z = q ) )
20	19	alrimivv 1782	. . 3 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> A. z A. q ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) -> z = q ) )
21		opeq12 4160	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> <. w , v >. = <. s , f >. )
22	21	eceq1d 7418	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
23	22	eqeq2d 2481	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> ( A = [ <. w , v >. ] ~R <-> A = [ <. s , f >. ] ~R ) )
24	23	anbi1d 719	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) ) )
25		simpl 464	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> w = s )
26	25	oveq1d 6323	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( w +P. u ) = ( s +P. u ) )
27		simpr 468	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> v = f )
28	27	oveq1d 6323	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( v +P. t ) = ( f +P. t ) )
29	26, 28	opeq12d 4166	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( s +P. u ) , ( f +P. t ) >. )
30	29	eceq1d 7418	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R )
31	30	eqeq2d 2481	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> q = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R ) )
32	24, 31	anbi12d 725	. . . . . . 7 |- ( ( w = s /\ v = f ) -> ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R ) ) )
33		opeq12 4160	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> <. u , t >. = <. g , h >. )
34	33	eceq1d 7418	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
35	34	eqeq2d 2481	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> ( B = [ <. u , t >. ] ~R <-> B = [ <. g , h >. ] ~R ) )
36	35	anbi2d 718	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) )
37		simpl 464	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> u = g )
38	37	oveq2d 6324	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( s +P. u ) = ( s +P. g ) )
39		simpr 468	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> t = h )
40	39	oveq2d 6324	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( f +P. t ) = ( f +P. h ) )
41	38, 40	opeq12d 4166	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> <. ( s +P. u ) , ( f +P. t ) >. = <. ( s +P. g ) , ( f +P. h ) >. )
42	41	eceq1d 7418	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
43	42	eqeq2d 2481	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( q = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R <-> q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) )
44	36, 43	anbi12d 725	. . . . . . 7 |- ( ( u = g /\ t = h ) -> ( ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R ) <-> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) )
45	32, 44	cbvex4v 2139	. . . . . 6 |- ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) )
46	45	anbi2i 708	. . . . 5 |- ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) <-> ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) )
47	46	imbi1i 332	. . . 4 |- ( ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> z = q ) <-> ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) -> z = q ) )
48	47	2albii 1700	. . 3 |- ( A. z A. q ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> z = q ) <-> A. z A. q ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R ) ) -> z = q ) )
49	20, 48	sylibr 217	. 2 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> A. z A. q ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> z = q ) )
50		eqeq1 2475	. . . . 5 |- ( z = q -> ( z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
51	50	anbi2d 718	. . . 4 |- ( z = q -> ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
52	51	4exbidv 1780	. . 3 |- ( z = q -> ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
53	52	mo4 2366	. 2 |- ( E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> A. z A. q ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) -> z = q ) )
54	49, 53	sylibr 217	1 |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   E*wmo 2320   <.cop 3965   class class class wbr 4395   X. cxp 4837  (class class class)co 6308   Er wer 7378   [cec 7379   /.cqs 7380   P.cnp 9302   +P. cpp 9304   ~R cer 9307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361  df-np 9424  df-plp 9426  df-ltp 9428  df-enr 9498

This theorem is referenced by:  addsrpr  9517