Theorem addsrmo 7855

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 7847	. . . . . . . . . . . . . . . 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 7854	. . . . . . . . . . . . . . . 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 7851	. . . . . . . . . . . . . . . . 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 124	. . . . . . . . . . . . . . . 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 6667	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 538	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 2247	. . . . . . . . . . 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 375	. . . . . . . . . 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 1920	. . . . . . . . 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 1920	. . . . . . . 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 115	. . . . . . 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 1920	. . . . . 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 1920	. . . . 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 254	. . . 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 1897	. . 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 3820	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> <. w , v >. = <. s , f >. )
22	21	eceq1d 6655	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
23	22	eqeq2d 2216	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> ( A = [ <. w , v >. ] ~R <-> A = [ <. s , f >. ] ~R ) )
24	23	anbi1d 465	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) ) )
25		simpl 109	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> w = s )
26	25	oveq1d 5958	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( w +P. u ) = ( s +P. u ) )
27		simpr 110	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> v = f )
28	27	oveq1d 5958	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( v +P. t ) = ( f +P. t ) )
29	26, 28	opeq12d 3826	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( s +P. u ) , ( f +P. t ) >. )
30	29	eceq1d 6655	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R )
31	30	eqeq2d 2216	. . . . . . . 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 473	. . . . . . 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 3820	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> <. u , t >. = <. g , h >. )
34	33	eceq1d 6655	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
35	34	eqeq2d 2216	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> ( B = [ <. u , t >. ] ~R <-> B = [ <. g , h >. ] ~R ) )
36	35	anbi2d 464	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) )
37		simpl 109	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> u = g )
38	37	oveq2d 5959	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( s +P. u ) = ( s +P. g ) )
39		simpr 110	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> t = h )
40	39	oveq2d 5959	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( f +P. t ) = ( f +P. h ) )
41	38, 40	opeq12d 3826	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> <. ( s +P. u ) , ( f +P. t ) >. = <. ( s +P. g ) , ( f +P. h ) >. )
42	41	eceq1d 6655	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> [ <. ( s +P. u ) , ( f +P. t ) >. ] ~R = [ <. ( s +P. g ) , ( f +P. h ) >. ] ~R )
43	42	eqeq2d 2216	. . . . . . . 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 473	. . . . . . 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 1957	. . . . . 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 457	. . . . 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 238	. . . 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 1493	. . 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 134	. 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 2211	. . . . 5 |- ( z = q -> ( z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> q = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
51	50	anbi2d 464	. . . 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 1892	. . 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 2114	. 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 134	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 104   A.wal 1370   = wceq 1372   E.wex 1514   E*wmo 2054   e. wcel 2175   <.cop 3635   class class class wbr 4043   X. cxp 4672  (class class class)co 5943   Er wer 6616   [cec 6617   /.cqs 6618   P.cnp 7403   +P. cpp 7405   ~R cer 7408

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iplp 7580  df-enr 7838

This theorem is referenced by:  addsrpr  7857