Theorem mulsrpr 3979

Description: Multiplication of signed reals in terms of positive reals.

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

Step	Hyp	Ref	Expression
1		opex 1893	. 2 |- <.((A .P C) +P. (B .P D)), ((A .P D) +P. (B .P C))>. e. V
2		opex 1893	. 2 |- <.((a .P g) +P. (b .P h)), ((a .P h) +P. (b .P g))>. e. V
3		opex 1893	. 2 |- <.((c .P t) +P. (d .P s)), ((c .P s) +P. (d .P t))>. e. V
4		enrex 3972	. 2 |- ~R e. V
5		enrer 3970	. 2 |- Er ~R
6		dmenr 3969	. 2 |- dom ~R = (P. X. P.)
7		df-enr 3960	. 2 |- ~R = {<.x, y>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (z +P. u) = (w +P. v)))}
8		opreq12 3008	. . . 4 |- ((z = a /\ u = d) -> (z +P. u) = (a +P. d))
9		opreq12 3008	. . . 4 |- ((w = b /\ v = c) -> (w +P. v) = (b +P. c))
10	8, 9	cleqan12d 1116	. . 3 |- (((z = a /\ u = d) /\ (w = b /\ v = c)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
11	10	an42s 391	. 2 |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
12		opreq12 3008	. . . 4 |- ((z = g /\ u = s) -> (z +P. u) = (g +P. s))
13		opreq12 3008	. . . 4 |- ((w = h /\ v = t) -> (w +P. v) = (h +P. t))
14	12, 13	cleqan12d 1116	. . 3 |- (((z = g /\ u = s) /\ (w = h /\ v = t)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
15	14	an42s 391	. 2 |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
16		df-mpr 3959	. 2 |- .pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>.))}
17		opeq12 1878	. . 3 |- ((((w .P u) +P. (v .P f)) = ((a .P g) +P. (b .P h)) /\ ((w .P f) +P. (v .P u)) = ((a .P h) +P. (b .P g))) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((a .P g) +P. (b .P h)), ((a .P h) +P. (b .P g))>.)
18		opreq12 3008	. . . . 5 |- ((w = a /\ u = g) -> (w .P u) = (a .P g))
19		opreq12 3008	. . . . 5 |- ((v = b /\ f = h) -> (v .P f) = (b .P h))
20	18, 19	opreqan12d 3015	. . . 4 |- (((w = a /\ u = g) /\ (v = b /\ f = h)) -> ((w .P u) +P. (v .P f)) = ((a .P g) +P. (b .P h)))
21	20	an4s 390	. . 3 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> ((w .P u) +P. (v .P f)) = ((a .P g) +P. (b .P h)))
22		opreq12 3008	. . . . 5 |- ((w = a /\ f = h) -> (w .P f) = (a .P h))
23		opreq12 3008	. . . . 5 |- ((v = b /\ u = g) -> (v .P u) = (b .P g))
24	22, 23	opreqan12d 3015	. . . 4 |- (((w = a /\ f = h) /\ (v = b /\ u = g)) -> ((w .P f) +P. (v .P u)) = ((a .P h) +P. (b .P g)))
25	24	an42s 391	. . 3 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> ((w .P f) +P. (v .P u)) = ((a .P h) +P. (b .P g)))
26	17, 21, 25	sylanc 361	. 2 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((a .P g) +P. (b .P h)), ((a .P h) +P. (b .P g))>.)
27		opeq12 1878	. . 3 |- ((((w .P u) +P. (v .P f)) = ((c .P t) +P. (d .P s)) /\ ((w .P f) +P. (v .P u)) = ((c .P s) +P. (d .P t))) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((c .P t) +P. (d .P s)), ((c .P s) +P. (d .P t))>.)
28		opreq12 3008	. . . . 5 |- ((w = c /\ u = t) -> (w .P u) = (c .P t))
29		opreq12 3008	. . . . 5 |- ((v = d /\ f = s) -> (v .P f) = (d .P s))
30	28, 29	opreqan12d 3015	. . . 4 |- (((w = c /\ u = t) /\ (v = d /\ f = s)) -> ((w .P u) +P. (v .P f)) = ((c .P t) +P. (d .P s)))
31	30	an4s 390	. . 3 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> ((w .P u) +P. (v .P f)) = ((c .P t) +P. (d .P s)))
32		opreq12 3008	. . . . 5 |- ((w = c /\ f = s) -> (w .P f) = (c .P s))
33		opreq12 3008	. . . . 5 |- ((v = d /\ u = t) -> (v .P u) = (d .P t))
34	32, 33	opreqan12d 3015	. . . 4 |- (((w = c /\ f = s) /\ (v = d /\ u = t)) -> ((w .P f) +P. (v .P u)) = ((c .P s) +P. (d .P t)))
35	34	an42s 391	. . 3 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> ((w .P f) +P. (v .P u)) = ((c .P s) +P. (d .P t)))
36	27, 31, 35	sylanc 361	. 2 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((c .P t) +P. (d .P s)), ((c .P s) +P. (d .P t))>.)
37		opeq12 1878	. . 3 |- ((((w .P u) +P. (v .P f)) = ((A .P C) +P. (B .P D)) /\ ((w .P f) +P. (v .P u)) = ((A .P D) +P. (B .P C))) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((A .P C) +P. (B .P D)), ((A .P D) +P. (B .P C))>.)
38		opreq12 3008	. . . . 5 |- ((w = A /\ u = C) -> (w .P u) = (A .P C))
39		opreq12 3008	. . . . 5 |- ((v = B /\ f = D) -> (v .P f) = (B .P D))
40	38, 39	opreqan12d 3015	. . . 4 |- (((w = A /\ u = C) /\ (v = B /\ f = D)) -> ((w .P u) +P. (v .P f)) = ((A .P C) +P. (B .P D)))
41	40	an4s 390	. . 3 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> ((w .P u) +P. (v .P f)) = ((A .P C) +P. (B .P D)))
42		opreq12 3008	. . . . 5 |- ((w = A /\ f = D) -> (w .P f) = (A .P D))
43		opreq12 3008	. . . . 5 |- ((v = B /\ u = C) -> (v .P u) = (B .P C))
44	42, 43	opreqan12d 3015	. . . 4 |- (((w = A /\ f = D) /\ (v = B /\ u = C)) -> ((w .P f) +P. (v .P u)) = ((A .P D) +P. (B .P C)))
45	44	an42s 391	. . 3 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> ((w .P f) +P. (v .P u)) = ((A .P D) +P. (B .P C)))
46	37, 41, 45	sylanc 361	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.((w .P u) +P. (v .P f)), ((w .P f) +P. (v .P u))>. = <.((