Theorem mulsrpr 5197

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 2788	. 2 |- <.((A .P. C) +P. (B .P. D)), ((A .P. D) +P. (B .P. C))>. e. V
2		opex 2788	. 2 |- <.((a .P. g) +P. (b .P. h)), ((a .P. h) +P. (b .P. g))>. e. V
3		opex 2788	. 2 |- <.((c .P. t) +P. (d .P. s)), ((c .P. s) +P. (d .P. t))>. e. V
4		enrex 5190	. 2 |- ~R e. V
5		enrer 5188	. 2 |- Er ~R
6		dmenr 5187	. 2 |- dom ~R = (P. X. P.)
7		df-enr 5178	. 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 3976	. . . 4 |- ((z = a /\ u = d) -> (z +P. u) = (a +P. d))
9		opreq12 3976	. . . 4 |- ((w = b /\ v = c) -> (w +P. v) = (b +P. c))
10	8, 9	eqeqan12d 1493	. . 3 |- (((z = a /\ u = d) /\ (w = b /\ v = c)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
11	10	an42s 511	. 2 |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
12		opreq12 3976	. . . 4 |- ((z = g /\ u = s) -> (z +P. u) = (g +P. s))
13		opreq12 3976	. . . 4 |- ((w = h /\ v = t) -> (w +P. v) = (h +P. t))
14	12, 13	eqeqan12d 1493	. . 3 |- (((z = g /\ u = s) /\ (w = h /\ v = t)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
15	14	an42s 511	. 2 |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
16		df-mpr 5177	. 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		opreq12 3976	. . . . 5 |- ((w = a /\ u = g) -> (w .P. u) = (a .P. g))
18		opreq12 3976	. . . . 5 |- ((v = b /\ f = h) -> (v .P. f) = (b .P. h))
19	17, 18	opreqan12d 3985	. . . 4 |- (((w = a /\ u = g) /\ (v = b /\ f = h)) -> ((w .P. u) +P. (v .P. f)) = ((a .P. g) +P. (b .P. h)))
20	19	an4s 510	. . 3 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> ((w .P. u) +P. (v .P. f)) = ((a .P. g) +P. (b .P. h)))
21		opreq12 3976	. . . . 5 |- ((w = a /\ f = h) -> (w .P. f) = (a .P. h))
22		opreq12 3976	. . . . 5 |- ((v = b /\ u = g) -> (v .P. u) = (b .P. g))
23	21, 22	opreqan12d 3985	. . . 4 |- (((w = a /\ f = h) /\ (v = b /\ u = g)) -> ((w .P. f) +P. (v .P. u)) = ((a .P. h) +P. (b .P. g)))
24	23	an42s 511	. . 3 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> ((w .P. f) +P. (v .P. u)) = ((a .P. h) +P. (b .P. g)))
25	20, 24	opeq12d 2499	. 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))>.)
26		opreq12 3976	. . . . 5 |- ((w = c /\ u = t) -> (w .P. u) = (c .P. t))
27		opreq12 3976	. . . . 5 |- ((v = d /\ f = s) -> (v .P. f) = (d .P. s))
28	26, 27	opreqan12d 3985	. . . 4 |- (((w = c /\ u = t) /\ (v = d /\ f = s)) -> ((w .P. u) +P. (v .P. f)) = ((c .P. t) +P. (d .P. s)))
29	28	an4s 510	. . 3 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> ((w .P. u) +P. (v .P. f)) = ((c .P. t) +P. (d .P. s)))
30		opreq12 3976	. . . . 5 |- ((w = c /\ f = s) -> (w .P. f) = (c .P. s))
31		opreq12 3976	. . . . 5 |- ((v = d /\ u = t) -> (v .P. u) = (d .P. t))
32	30, 31	opreqan12d 3985	. . . 4 |- (((w = c /\ f = s) /\ (v = d /\ u = t)) -> ((w .P. f) +P. (v .P. u)) = ((c .P. s) +P. (d .P. t)))
33	32	an42s 511	. . 3 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> ((w .P. f) +P. (v .P. u)) = ((c .P. s) +P. (d .P. t)))
34	29, 33	opeq12d 2499	. 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))>.)
35		opreq12 3976	. . . . 5 |- ((w = A /\ u = C) -> (w .P. u) = (A .P. C))
36		opreq12 3976	. . . . 5 |- ((v = B /\ f = D) -> (v .P. f) = (B .P. D))
37	35, 36	opreqan12d 3985	. . . 4 |- (((w = A /\ u = C) /\ (v = B /\ f = D)) -> ((w .P. u) +P. (v .P. f)) = ((A .P. C) +P. (B .P. D)))
38	37	an4s 510	. . 3 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> ((w .P. u) +P. (v .P. f)) = ((A .P. C) +P. (B .P. D)))
39		opreq12 3976	. . . . 5 |- ((w = A /\ f = D) -> (w .P. f) = (A .P. D))
40		opreq12 3976	. . . . 5 |- ((v = B /\ u = C) -> (v .P. u) = (B .P. C))
41	39, 40	opreqan12d 3985	. . . 4 |- (((w = A /\ f = D) /\ (v = B /\ u = C)) -> ((w .P. f) +P. (v .P. u)) = ((A .P. D) +P. (B .P. C)))
42	41	an42s 511	. . 3 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> ((w .P. f) +P. (v .P. u)) = ((A .P. D) +P. (B .P. C)))
43	38, 42	opeq12d 2499	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.((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))>.)
44		df-mr 5181	. 2 |- .R = {<.<.x, y>., z>. | ((x e. R. /\ y e. R.) /\ E.aE.bE.cE.d((x = [<.a, b>.] ~R /\ y = [<.c, d>.] ~R ) /\ z = [(<.a, b>. .pR <.c, d>.)] ~R ))}
45		df-nr 5179	. 2 |- R. = ((P. X. P.)/. ~R )
46		visset 1816	. . 3 |- a e. V
47		visset 1816	. . 3 |- b e. V
48		visset 1816	. . 3 |- c e. V
49		visset 1816	. . 3 |- d e. V
50		visset 1816	. . 3 |- g e. V
51		visset 1816	. . 3 |- h e. V
52		visset 1816	. . 3 |- t e. V
53		visset 1816	. . 3 |- s e. V
54	46, 47, 48, 49, 50, 51, 52, 53	mulcmpblnr 5195	. 2 |- ((((a e. P. /\ b e. P.) /\ (c e. P. /\ d e. P.)) /\ ((g e. P. /\ h e. P.) /\ (t e. P. /\ s e. P.))) -> (((a +P. d) = (b +P. c) /\ (g +P. s) = (h +P. t)) -> <.((a .P. g) +P. (b .P. h)), ((a .P. h) +P. (b .P. g))>. ~R <.((c .P. t) +P. (d .P. s))