Theorem addsrmo 7663

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 7655	. . . . . . . . . . . . . . . 16 |- ~R Er ( P. X. P. )
2	1	a1i 9	. . . . . . . . . . . . . . 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 7662	. . . . . . . . . . . . . . . 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 7659	. . . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . 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 6526	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 528	. . . . . . . . . . . 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 530	. . . . . . . . . . . 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 2200	. . . . . . . . . . 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 373	. . . . . . . . . 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 1877	. . . . . . . . 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 1877	. . . . . . . 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 114	. . . . . . 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 1877	. . . . . 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 1877	. . . . 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 252	. . . 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 1855	. . 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 3743	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> <. w , v >. = <. s , f >. )
22	21	eceq1d 6516	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
23	22	eqeq2d 2169	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> ( A = [ <. w , v >. ] ~R <-> A = [ <. s , f >. ] ~R ) )
24	23	anbi1d 461	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) ) )
25		simpl 108	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> w = s )
26	25	oveq1d 5839	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( w +P. u ) = ( s +P. u ) )
27		simpr 109	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> v = f )
28	27	oveq1d 5839	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( v +P. t ) = ( f +P. t ) )
29	26, 28	opeq12d 3749	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( s +P. u ) , ( f +P. t ) >. )
30	29	eceq1d 6516	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R )
31	30	eqeq2d 2169	. . . . . . . 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 465	. . . . . . 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 3743	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> <. u , t >. = <. g , h >. )
34	33	eceq1d 6516	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
35	34	eqeq2d 2169	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> ( B = [ <. u , t >. ] ~R <-> B = [ <. g , h >. ] ~R ) )
36	35	anbi2d 460	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) )
37		simpl 108	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> u = g )
38	37	oveq2d 5840	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( s +P. u ) = ( s +P. g ) )
39		simpr 109	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> t = h )
40	39	oveq2d 5840	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( f +P. t ) = ( f +P. h ) )
41	38, 40	opeq12d 3749	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> <. ( s +P. u ) , ( f +P. t ) >. = <. ( s +P. g ) , ( f +P. h ) >. )
42	41	eceq1d 6516	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
43	42	eqeq2d 2169	. . . . . . . 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 465	. . . . . . 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 1910	. . . . . 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 453	. . . . 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 237	. . . 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 1451	. . 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 133	. 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 2164	. . . . 5 |- ( z = q -> ( z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
51	50	anbi2d 460	. . . 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 1850	. . 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 2067	. 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 133	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 103   A.wal 1333   = wceq 1335   E.wex 1472   E*wmo 2007   e. wcel 2128   <.cop 3563   class class class wbr 3965   X. cxp 4584  (class class class)co 5824   Er wer 6477   [cec 6478   /.cqs 6479   P.cnp 7211   +P. cpp 7213   ~R cer 7216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-enr 7646

This theorem is referenced by:  addsrpr  7665