Theorem mulsrmo 7706

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 7697	. . . . . . . . . . . . . . . 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 7704	. . . . . . . . . . . . . . . 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 7703	. . . . . . . . . . . . . . . . 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 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 ) +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 6559	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 533	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 2213	. . . . . . . . . . 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 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 ) +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 1890	. . . . . . . . 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 1890	. . . . . . . 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 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 ) +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 1890	. . . . . 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 1890	. . . . 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 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 ) +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 1868	. . 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 3767	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> <. w , v >. = <. s , f >. )
22	21	eceq1d 6549	. . . . . . . . . 10 |- ( ( w = s /\ v = f ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
23	22	eqeq2d 2182	. . . . . . . . 9 |- ( ( w = s /\ v = f ) -> ( A = [ <. w , v >. ] ~R <-> A = [ <. s , f >. ] ~R ) )
24	23	anbi1d 462	. . . . . . . 8 |- ( ( w = s /\ v = f ) -> ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) ) )
25		simpl 108	. . . . . . . . . . . . 13 |- ( ( w = s /\ v = f ) -> w = s )
26	25	oveq1d 5868	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( w .P. u ) = ( s .P. u ) )
27		simpr 109	. . . . . . . . . . . . 13 |- ( ( w = s /\ v = f ) -> v = f )
28	27	oveq1d 5868	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( v .P. t ) = ( f .P. t ) )
29	26, 28	oveq12d 5871	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( s .P. u ) +P. ( f .P. t ) ) )
30	25	oveq1d 5868	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( w .P. t ) = ( s .P. t ) )
31	27	oveq1d 5868	. . . . . . . . . . . 12 |- ( ( w = s /\ v = f ) -> ( v .P. u ) = ( f .P. u ) )
32	30, 31	oveq12d 5871	. . . . . . . . . . 11 |- ( ( w = s /\ v = f ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( s .P. t ) +P. ( f .P. u ) ) )
33	29, 32	opeq12d 3773	. . . . . . . . . 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 6549	. . . . . . . . 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 2182	. . . . . . . 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 470	. . . . . . 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 3767	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> <. u , t >. = <. g , h >. )
38	37	eceq1d 6549	. . . . . . . . . 10 |- ( ( u = g /\ t = h ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
39	38	eqeq2d 2182	. . . . . . . . 9 |- ( ( u = g /\ t = h ) -> ( B = [ <. u , t >. ] ~R <-> B = [ <. g , h >. ] ~R ) )
40	39	anbi2d 461	. . . . . . . 8 |- ( ( u = g /\ t = h ) -> ( ( A = [ <. s , f >. ] ~R /\ B = [ <. u , t >. ] ~R ) <-> ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) )
41		simpl 108	. . . . . . . . . . . . 13 |- ( ( u = g /\ t = h ) -> u = g )
42	41	oveq2d 5869	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( s .P. u ) = ( s .P. g ) )
43		simpr 109	. . . . . . . . . . . . 13 |- ( ( u = g /\ t = h ) -> t = h )
44	43	oveq2d 5869	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( f .P. t ) = ( f .P. h ) )
45	42, 44	oveq12d 5871	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( ( s .P. u ) +P. ( f .P. t ) ) = ( ( s .P. g ) +P. ( f .P. h ) ) )
46	43	oveq2d 5869	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( s .P. t ) = ( s .P. h ) )
47	41	oveq2d 5869	. . . . . . . . . . . 12 |- ( ( u = g /\ t = h ) -> ( f .P. u ) = ( f .P. g ) )
48	46, 47	oveq12d 5871	. . . . . . . . . . 11 |- ( ( u = g /\ t = h ) -> ( ( s .P. t ) +P. ( f .P. u ) ) = ( ( s .P. h ) +P. ( f .P. g ) ) )
49	45, 48	opeq12d 3773	. . . . . . . . . 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 6549	. . . . . . . . 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 2182	. . . . . . . 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 470	. . . . . . 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 1923	. . . . . 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 454	. . . . 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 237	. . . 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 1464	. . 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 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 ) +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 2177	. . . . 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 461	. . . 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 1863	. . 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 2080	. 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 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 ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   <.cop 3586   class class class wbr 3989   X. cxp 4609  (class class class)co 5853   Er wer 6510   [cec 6511   /.cqs 6512   P.cnp 7253   +P. cpp 7255   .P. cmp 7256   ~R cer 7258

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431  df-enr 7688

This theorem is referenced by:  mulsrpr  7708