Theorem mulsrpr 8026

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 4763	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 8017	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6801	. . . 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 4763	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6801	. . . 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 2231	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2231	. . . 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 2231	. . 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 3869	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6781	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2243	. . . . . . 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 6043	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. C ) = ( A .P. C ) )
19		simpr 110	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 6043	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. D ) = ( B .P. D ) )
21	18, 20	oveq12d 6046	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. C ) +P. ( v .P. D ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
22	17	oveq1d 6043	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. D ) = ( A .P. D ) )
23	19	oveq1d 6043	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. C ) = ( B .P. C ) )
24	22, 23	oveq12d 6046	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. D ) +P. ( v .P. C ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
25	21, 24	opeq12d 3875	. . . . . . . 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 6781	. . . . . . 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 2243	. . . . . 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 2897	. . . 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 3869	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
31	30	eceq1d 6781	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
32	31	eqeq2d 2243	. . . . . . . 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 6044	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. u ) = ( w .P. C ) )
36		simpr 110	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> t = D )
37	36	oveq2d 6044	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. t ) = ( v .P. D ) )
38	35, 37	oveq12d 6046	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( w .P. C ) +P. ( v .P. D ) ) )
39	36	oveq2d 6044	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. t ) = ( w .P. D ) )
40	34	oveq2d 6044	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. u ) = ( v .P. C ) )
41	39, 40	oveq12d 6046	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( w .P. D ) +P. ( v .P. C ) ) )
42	38, 41	opeq12d 3875	. . . . . . . . 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 6781	. . . . . . . 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 2243	. . . . . . 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 2897	. . . . 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 1930	. . . 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 6749	. . . 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 1024	. . . . . . . 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 2240	. . . . . . 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 1025	. . . . . . . 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 2240	. . . . . . 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 1026	. . . . . . 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 2240	. . . . . 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 1918	. . . 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 8024	. . . 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 8009	. . . . 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 8007	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
64	63	eleq2i 2298	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
65	63	eleq2i 2298	. . . . . . . 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 6086	. . . . 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 2252	. . . 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 6153	. . 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 1363	. 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 1005   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   <.cop 3676   X. cxp 4729  (class class class)co 6028   {coprab 6029   [cec 6743   /.cqs 6744   P.cnp 7571   +P. cpp 7573   .P. cmp 7574   ~R cer 7576   R.cnr 7577   .R cmr 7582

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-iplp 7748  df-imp 7749  df-enr 8006  df-nr 8007  df-mr 8009

This theorem is referenced by:  mulclsr  8034  mulcomsrg  8037  mulasssrg  8038  distrsrg  8039  m1m1sr  8041  1idsr  8048  00sr  8049  recexgt0sr  8053  mulgt0sr  8058  mulextsr1  8061  recidpirq  8138