Theorem mulsrpr 8106

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 4804	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 8097	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6856	. . . 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 4804	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6856	. . . 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 2238	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2238	. . . 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 2238	. . 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 3904	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6836	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2250	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. A , B >. ] ~R ) )
16	15	anbi1d 469	. . . . . 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 6093	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. C ) = ( A .P. C ) )
19		simpr 110	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 6093	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. D ) = ( B .P. D ) )
21	18, 20	oveq12d 6096	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. C ) +P. ( v .P. D ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
22	17	oveq1d 6093	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. D ) = ( A .P. D ) )
23	19	oveq1d 6093	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. C ) = ( B .P. C ) )
24	22, 23	oveq12d 6096	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. D ) +P. ( v .P. C ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
25	21, 24	opeq12d 3910	. . . . . . . 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 6836	. . . . . . 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 2250	. . . . . 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 477	. . . . 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 2915	. . . 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 3904	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
31	30	eceq1d 6836	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
32	31	eqeq2d 2250	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~R = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) )
33	32	anbi2d 468	. . . . . . 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 6094	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. u ) = ( w .P. C ) )
36		simpr 110	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> t = D )
37	36	oveq2d 6094	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. t ) = ( v .P. D ) )
38	35, 37	oveq12d 6096	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( w .P. C ) +P. ( v .P. D ) ) )
39	36	oveq2d 6094	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. t ) = ( w .P. D ) )
40	34	oveq2d 6094	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. u ) = ( v .P. C ) )
41	39, 40	oveq12d 6096	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( w .P. D ) +P. ( v .P. C ) ) )
42	38, 41	opeq12d 3910	. . . . . . . . 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 6836	. . . . . . . 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 2250	. . . . . . 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 477	. . . . . 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 2915	. . . . 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 1935	. . . 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 413	. . 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 436	. 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 6804	. . . 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 1028	. . . . . . . 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 2247	. . . . . . 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 1029	. . . . . . . 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 2247	. . . . . . 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 477	. . . . . 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 1030	. . . . . . 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 2247	. . . . . 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 477	. . . . 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 1923	. . . 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 8104	. . . 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 8089	. . . . 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 8087	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
64	63	eleq2i 2305	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
65	63	eleq2i 2305	. . . . . . . 8 |- ( y e. R. <-> y e. ( ( P. X. P. ) /. ~R ) )
66	64, 65	anbi12i 464	. . . . . . 7 |- ( ( x e. R. /\ y e. R. ) <-> ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) )
67	66	anbi1i 462	. . . . . 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 6136	. . . . 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 2259	. . . 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 6203	. . 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 1367	. 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 1009   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3711   X. cxp 4770  (class class class)co 6078   {coprab 6079   [cec 6798   /.cqs 6799   P.cnp 7651   +P. cpp 7653   .P. cmp 7654   ~R cer 7656   R.cnr 7657   .R cmr 7662

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-irdg 6634  df-1o 6680  df-2o 6681  df-oadd 6684  df-omul 6685  df-er 6800  df-ec 6802  df-qs 6806  df-ni 7664  df-pli 7665  df-mi 7666  df-lti 7667  df-plpq 7704  df-mpq 7705  df-enq 7707  df-nqqs 7708  df-plqqs 7709  df-mqqs 7710  df-1nqqs 7711  df-rq 7712  df-ltnqqs 7713  df-enq0 7784  df-nq0 7785  df-0nq0 7786  df-plq0 7787  df-mq0 7788  df-inp 7826  df-iplp 7828  df-imp 7829  df-enr 8086  df-nr 8087  df-mr 8089

This theorem is referenced by:  mulclsr  8114  mulcomsrg  8117  mulasssrg  8118  distrsrg  8119  m1m1sr  8121  1idsr  8128  00sr  8129  recexgt0sr  8133  mulgt0sr  8138  mulextsr1  8141  recidpirq  8218