Theorem mulsrmo 6829

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

Assertion

Ref	Expression
mulsrmo	|- ( ( 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )

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

Proof of Theorem mulsrmo

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

Step	Hyp	Ref	Expression
1		enrer 6820	. . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . 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		mulcmpblnr 6826	. . . . . . . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ~R <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ) )
5	4	imp 115	. . . . . . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ~R <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. )
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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ~R <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. )
7	2, 6	erthi 6152	. . . . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R )
8	7	adantrlr 454	. . . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R )
9	8	adantrrr 456	. . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) ) -> [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R )
10		simprlr 490	. . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) ) -> z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R )
11		simprrr 492	. . . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) ) -> q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R )
12	9, 10, 11	3eqtr4d 2082	. . . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) ) -> z = q )
13	12	expr 357	. . . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> ( ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) )
14	13	exlimdvv 1777	. . . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> ( E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) )
15	14	exlimdvv 1777	. . . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) )
16	15	ex 108	. . . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) ) )
17	16	exlimdvv 1777	. . . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) ) )
18	17	exlimdvv 1777	. . . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) -> z = q ) ) )
19	18	impd 242	. . . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) -> z = q ) )
20	19	alrimivv 1755	. . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) -> z = q ) )
21		opeq12 3551	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> <. w , v >. = <. s , f >. )
22	21	eceq1d 6142	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
23	22	eqeq2d 2051	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> ( A = [ <. w , v >. ] ~R <-> A = [ <. s , f >. ] ~R ) )
24	23	anbi1d 438	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) ) )
25		simpl 102	. . . . . . . . . . . . 13 |- ( ( w = s /\ v = f ) -> w = s )
26	25	oveq1d 5527	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( w .P. u ) = ( s .P. u ) )
27		simpr 103	. . . . . . . . . . . . 13 |- ( ( w = s /\ v = f ) -> v = f )
28	27	oveq1d 5527	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( v .P. t ) = ( f .P. t ) )
29	26, 28	oveq12d 5530	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( s .P. u ) +P. ( f .P. t ) ) )
30	25	oveq1d 5527	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( w .P. t ) = ( s .P. t ) )
31	27	oveq1d 5527	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( v .P. u ) = ( f .P. u ) )
32	30, 31	oveq12d 5530	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( s .P. t ) +P. ( f .P. u ) ) )
33	29, 32	opeq12d 3557	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. = <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. )
34	33	eceq1d 6142	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R )
35	34	eqeq2d 2051	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> q = [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R ) )
36	24, 35	anbi12d 442	. . . . . . 7 |- ( ( w = s /\ v = f ) -> ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R ) ) )
37		opeq12 3551	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> <. u , t >. = <. g , h >. )
38	37	eceq1d 6142	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
39	38	eqeq2d 2051	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> ( B = [ <. u , t >. ] ~R <-> B = [ <. g , h >. ] ~R ) )
40	39	anbi2d 437	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) )
41		simpl 102	. . . . . . . . . . . . 13 |- ( ( u = g /\ t = h ) -> u = g )
42	41	oveq2d 5528	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( s .P. u ) = ( s .P. g ) )
43		simpr 103	. . . . . . . . . . . . 13 |- ( ( u = g /\ t = h ) -> t = h )
44	43	oveq2d 5528	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( f .P. t ) = ( f .P. h ) )
45	42, 44	oveq12d 5530	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( ( s .P. u ) +P. ( f .P. t ) ) = ( ( s .P. g ) +P. ( f .P. h ) ) )
46	43	oveq2d 5528	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( s .P. t ) = ( s .P. h ) )
47	41	oveq2d 5528	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( f .P. u ) = ( f .P. g ) )
48	46, 47	oveq12d 5530	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( ( s .P. t ) +P. ( f .P. u ) ) = ( ( s .P. h ) +P. ( f .P. g ) ) )
49	45, 48	opeq12d 3557	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. = <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. )
50	49	eceq1d 6142	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R )
51	50	eqeq2d 2051	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( q = [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R <-> q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) )
52	40, 51	anbi12d 442	. . . . . . 7 |- ( ( u = g /\ t = h ) -> ( ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( s .P. u ) +P. ( f .P. t ) ) , ( ( s .P. t ) +P. ( f .P. u ) ) >. ] ~R ) <-> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) )
53	36, 52	cbvex4v 1805	. . . . . 6 |- ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) )
54	53	anbi2i 430	. . . . 5 |- ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) <-> ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) )
55	54	imbi1i 227	. . . 4 |- ( ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> z = q ) <-> ( ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) -> z = q ) )
56	55	2albii 1360	. . 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. s E. f E. g E. h ( ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) /\ q = [ <. ( ( s .P. g ) +P. ( f .P. h ) ) , ( ( s .P. h ) +P. ( f .P. g ) ) >. ] ~R ) ) -> z = q ) )
57	20, 56	sylibr 137	. 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> z = q ) )
58		eqeq1 2046	. . . . 5 |- ( z = q -> ( z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
59	58	anbi2d 437	. . . 4 |- ( z = q -> ( ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
60	59	4exbidv 1750	. . 3 |- ( z = q -> ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
61	60	mo4 1961	. 2 |- ( E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) /\ E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ q = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) -> z = q ) )
62	57, 61	sylibr 137	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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E*wmo 1901   <.cop 3378   class class class wbr 3764   X. cxp 4343  (class class class)co 5512   Er wer 6103   [cec 6104   /.cqs 6105   P.cnp 6389   +P. cpp 6391   .P. cmp 6392   ~R cer 6394

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

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-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-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-iplp 6566  df-imp 6567  df-enr 6811

This theorem is referenced by:  mulsrpr  6831