Theorem mulsrpr 5339

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 2858	. 2 |- <.((A .P. C) +P. (B .P. D)), ((A .P. D) +P. (B .P. C))>. e. V
2		opex 2858	. 2 |- <.((a .P. g) +P. (b .P. h)), ((a .P. h) +P. (b .P. g))>. e. V
3		opex 2858	. 2 |- <.((c .P. t) +P. (d .P. s)), ((c .P. s) +P. (d .P. t))>. e. V
4		enrex 5332	. 2 |- ~R e. V
5		enrer 5330	. 2 |- Er ~R
6		dmenr 5329	. 2 |- dom ~R = (P. X. P.)
7		df-enr 5320	. 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 4028	. . . 4 |- ((z = a /\ u = d) -> (z +P. u) = (a +P. d))
9		opreq12 4028	. . . 4 |- ((w = b /\ v = c) -> (w +P. v) = (b +P. c))
10	8, 9	eqeqan12d 1533	. . 3 |- (((z = a /\ u = d) /\ (w = b /\ v = c)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
11	10	an42s 512	. 2 |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
12		opreq12 4028	. . . 4 |- ((z = g /\ u = s) -> (z +P. u) = (g +P. s))
13		opreq12 4028	. . . 4 |- ((w = h /\ v = t) -> (w +P. v) = (h +P. t))
14	12, 13	eqeqan12d 1533	. . 3 |- (((z = g /\ u = s) /\ (w = h /\ v = t)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
15	14	an42s 512	. 2 |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
16		df-mpr 5319	. 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 4028	. . . . 5 |- ((w = a /\ u = g) -> (w .P. u) = (a .P. g))
18		opreq12 4028	. . . . 5 |- ((v = b /\ f = h) -> (v .P. f) = (b .P. h))
19	17, 18	opreqan12d 4037	. . . 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 511	. . 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 4028	. . . . 5 |- ((w = a /\ f = h) -> (w .P. f) = (a .P. h))
22		opreq12 4028	. . . . 5 |- ((v = b /\ u = g) -> (v .P. u) = (b .P. g))
23	21, 22	opreqan12d 4037	. . . 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 512	. . 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 2560	. 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 4028	. . . . 5 |- ((w = c /\ u = t) -> (w .P. u) = (c .P. t))
27		opreq12 4028	. . . . 5 |- ((v = d /\ f = s) -> (v .P. f) = (d .P. s))
28	26, 27	opreqan12d 4037	. . . 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 511	. . 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 4028	. . . . 5 |- ((w = c /\ f = s) -> (w .P. f) = (c .P. s))
31		opreq12 4028	. . . . 5 |- ((v = d /\ u = t) -> (v .P. u) = (d .P. t))
32	30, 31	opreqan12d 4037	. . . 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 512	. . 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 2560	. 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 4028	. . . . 5 |- ((w = A /\ u = C) -> (w .P. u) = (A .P. C))
36		opreq12 4028	. . . . 5 |- ((v = B /\ f = D) -> (v .P. f) = (B .P. D))
37	35, 36	opreqan12d 4037	. . . 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 511	. . 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 4028	. . . . 5 |- ((w = A /\ f = D) -> (w .P. f) = (A .P. D))
40		opreq12 4028	. . . . 5 |- ((v = B /\ u = C) -> (v .P. u) = (B .P. C))
41	39, 40	opreqan12d 4037	. . . 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 512	. . 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 2560	. 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 5323	. 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 5321	. 2 |- R. = ((P. X. P.)/. ~R )
46		visset 1859	. . 3 |- a e. V
47		visset 1859	. . 3 |- b e. V
48		visset 1859	. . 3 |- c e. V
49		visset 1859	. . 3 |- d e. V
50		visset 1859	. . 3 |- g e. V
51		visset 1859	. . 3 |- h e. V
52		visset 1859	. . 3 |- t e. V
53		visset 1859	. . 3 |- s e. V
54	46, 47, 48, 49, 50, 51, 52, 53	mulcmpblnr 5337	. 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)), ((c .P. s) +P. (d .P. t))>.))
55	1, 2, 3, 4, 5, 6, 7, 11, 15, 16, 25, 34, 43, 44, 45, 54	oprec 4459	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 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  <.cop 2469  (class class class)co 4021  [cec 4399  P.cnp 5139   +P. cpp 5141   .P. cmp 5142   .pR cmpr 5145   ~R cer 5146  R.cnr 5147   .R cmr 5152

This theorem is referenced by:  mulclsr 5347  mulcomsr 5352  mulasssr 5353  distrsr 5354  m1m1sr 5356  1idsr 5361  00sr 5362  recexsrlem 5366  mulgt0sr 5368

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-plp 5242  df-mp 5243  df-ltp 5244  df-mpr 5319  df-enr 5320  df-nr 5321  df-mr 5323

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