Theorem mulsrpr 7813

Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
mulsrpr	|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Proof of Theorem mulsrpr

Dummy variables x y z w v u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 4695	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 7804	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6648	. . . 4 |- ( <. A , B >. e. ( P. X. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
4	1, 3	syl 14	. . 3 |- ( ( A e. P. /\ B e. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
5		opelxpi 4695	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6648	. . . 4 |- ( <. C , D >. e. ( P. X. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
7	5, 6	syl 14	. . 3 |- ( ( C e. P. /\ D e. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
8	4, 7	anim12i 338	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) )
9		eqid 2196	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2196	. . . 4 |- [ <. C , D >. ] ~R = [ <. C , D >. ] ~R
11	9, 10	pm3.2i 272	. . 3 |- ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R )
12		eqid 2196	. . 3 |- [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R
13		opeq12 3810	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6628	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2208	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. A , B >. ] ~R ) )
16	15	anbi1d 465	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
17		simpl 109	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> w = A )
18	17	oveq1d 5937	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. C ) = ( A .P. C ) )
19		simpr 110	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 5937	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. D ) = ( B .P. D ) )
21	18, 20	oveq12d 5940	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. C ) +P. ( v .P. D ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
22	17	oveq1d 5937	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. D ) = ( A .P. D ) )
23	19	oveq1d 5937	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. C ) = ( B .P. C ) )
24	22, 23	oveq12d 5940	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. D ) +P. ( v .P. C ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
25	21, 24	opeq12d 3816	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. = <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. )
26	25	eceq1d 6628	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
27	26	eqeq2d 2208	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
28	16, 27	anbi12d 473	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) ) )
29	28	spc2egv 2854	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) ) )
30		opeq12 3810	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
31	30	eceq1d 6628	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
32	31	eqeq2d 2208	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~R = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) )
33	32	anbi2d 464	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
34		simpl 109	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> u = C )
35	34	oveq2d 5938	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. u ) = ( w .P. C ) )
36		simpr 110	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> t = D )
37	36	oveq2d 5938	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. t ) = ( v .P. D ) )
38	35, 37	oveq12d 5940	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( w .P. C ) +P. ( v .P. D ) ) )
39	36	oveq2d 5938	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. t ) = ( w .P. D ) )
40	34	oveq2d 5938	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. u ) = ( v .P. C ) )
41	39, 40	oveq12d 5940	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( w .P. D ) +P. ( v .P. C ) ) )
42	38, 41	opeq12d 3816	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. = <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. )
43	42	eceq1d 6628	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R )
44	43	eqeq2d 2208	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) )
45	33, 44	anbi12d 473	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) ) )
46	45	spc2egv 2854	. . . . 5 |- ( ( C e. P. /\ D e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) -> E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
47	46	2eximdv 1896	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> ( E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
48	29, 47	sylan9 409	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
49	11, 12, 48	mp2ani 432	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
50		ecexg 6596	. . . 4 |- ( ~R e. _V -> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V )
51	2, 50	ax-mp 5	. . 3 |- [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V
52		simp1 999	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> x = [ <. A , B >. ] ~R )
53	52	eqeq1d 2205	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( x = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. w , v >. ] ~R ) )
54		simp2 1000	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> y = [ <. C , D >. ] ~R )
55	54	eqeq1d 2205	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( y = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) )
56	53, 55	anbi12d 473	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) ) )
57		simp3 1001	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
58	57	eqeq1d 2205	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
59	56, 58	anbi12d 473	. . . . 5 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
60	59	4exbidv 1884	. . . 4 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. 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 , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
61		mulsrmo 7811	. . . 4 |- ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
62		df-mr 7796	. . . . 5 |- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
63		df-nr 7794	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
64	63	eleq2i 2263	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
65	63	eleq2i 2263	. . . . . . . 8 |- ( y e. R. <-> y e. ( ( P. X. P. ) /. ~R ) )
66	64, 65	anbi12i 460	. . . . . . 7 |- ( ( x e. R. /\ y e. R. ) <-> ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) )
67	66	anbi1i 458	. . . . . 6 |- ( ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) <-> ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
68	67	oprabbii 5977	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
69	62, 68	eqtri 2217	. . . 4 |- .R = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
70	60, 61, 69	ovig 6044	. . 3 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
71	51, 70	mp3an3 1337	. 2 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
72	8, 49, 71	sylc 62	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   <.cop 3625   X. cxp 4661  (class class class)co 5922   {coprab 5923   [cec 6590   /.cqs 6591   P.cnp 7358   +P. cpp 7360   .P. cmp 7361   ~R cer 7363   R.cnr 7364   .R cmr 7369

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-iplp 7535  df-imp 7536  df-enr 7793  df-nr 7794  df-mr 7796

This theorem is referenced by:  mulclsr  7821  mulcomsrg  7824  mulasssrg  7825  distrsrg  7826  m1m1sr  7828  1idsr  7835  00sr  7836  recexgt0sr  7840  mulgt0sr  7845  mulextsr1  7848  recidpirq  7925