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Definition df-mr 6637
Description: Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
Assertion
Ref Expression
df-mr ·R = {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R ))}
Distinct variable group:   x,y,z,w,v,u,f

Detailed syntax breakdown of Definition df-mr
StepHypRef Expression
1 cmr 6286 . 2 class ·R
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cnr 6281 . . . . . 6 class R
53, 4wcel 1390 . . . . 5 wff x R
6 vy . . . . . . 7 setvar y
76cv 1241 . . . . . 6 class y
87, 4wcel 1390 . . . . 5 wff y R
95, 8wa 97 . . . 4 wff (x R y R)
10 vw . . . . . . . . . . . . . 14 setvar w
1110cv 1241 . . . . . . . . . . . . 13 class w
12 vv . . . . . . . . . . . . . 14 setvar v
1312cv 1241 . . . . . . . . . . . . 13 class v
1411, 13cop 3370 . . . . . . . . . . . 12 class w, v
15 cer 6280 . . . . . . . . . . . 12 class ~R
1614, 15cec 6040 . . . . . . . . . . 11 class [⟨w, v⟩] ~R
173, 16wceq 1242 . . . . . . . . . 10 wff x = [⟨w, v⟩] ~R
18 vu . . . . . . . . . . . . . 14 setvar u
1918cv 1241 . . . . . . . . . . . . 13 class u
20 vf . . . . . . . . . . . . . 14 setvar f
2120cv 1241 . . . . . . . . . . . . 13 class f
2219, 21cop 3370 . . . . . . . . . . . 12 class u, f
2322, 15cec 6040 . . . . . . . . . . 11 class [⟨u, f⟩] ~R
247, 23wceq 1242 . . . . . . . . . 10 wff y = [⟨u, f⟩] ~R
2517, 24wa 97 . . . . . . . . 9 wff (x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R )
26 vz . . . . . . . . . . 11 setvar z
2726cv 1241 . . . . . . . . . 10 class z
28 cmp 6278 . . . . . . . . . . . . . 14 class ·P
2911, 19, 28co 5455 . . . . . . . . . . . . 13 class (w ·P u)
3013, 21, 28co 5455 . . . . . . . . . . . . 13 class (v ·P f)
31 cpp 6277 . . . . . . . . . . . . 13 class +P
3229, 30, 31co 5455 . . . . . . . . . . . 12 class ((w ·P u) +P (v ·P f))
3311, 21, 28co 5455 . . . . . . . . . . . . 13 class (w ·P f)
3413, 19, 28co 5455 . . . . . . . . . . . . 13 class (v ·P u)
3533, 34, 31co 5455 . . . . . . . . . . . 12 class ((w ·P f) +P (v ·P u))
3632, 35cop 3370 . . . . . . . . . . 11 class ⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩
3736, 15cec 6040 . . . . . . . . . 10 class [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R
3827, 37wceq 1242 . . . . . . . . 9 wff z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R
3925, 38wa 97 . . . . . . . 8 wff ((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R )
4039, 20wex 1378 . . . . . . 7 wff f((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R )
4140, 18wex 1378 . . . . . 6 wff uf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R )
4241, 12wex 1378 . . . . 5 wff vuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R )
4342, 10wex 1378 . . . 4 wff wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R )
449, 43wa 97 . . 3 wff ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R ))
4544, 2, 6, 26coprab 5456 . 2 class {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R ))}
461, 45wceq 1242 1 wff ·R = {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R ))}
Colors of variables: wff set class
This definition is referenced by:  mulsrpr  6654
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