Definition df-mr 9759

Description: Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9821, 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.) (New usage is discouraged.)

Assertion

Ref	Expression
df-mr	⊢ ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃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 ))}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-mr

Step	Hyp	Ref	Expression
1		cmr 9571	. 2 class ·R
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1474	. . . . . 6 class 𝑥
4		cnr 9566	. . . . . 6 class R
5	3, 4	wcel 1977	. . . . 5 wff 𝑥 ∈ R
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1474	. . . . . 6 class 𝑦
8	7, 4	wcel 1977	. . . . 5 wff 𝑦 ∈ R
9	5, 8	wa 383	. . . 4 wff (𝑥 ∈ R ∧ 𝑦 ∈ R)
10		vw	. . . . . . . . . . . . . 14 setvar 𝑤
11	10	cv 1474	. . . . . . . . . . . . 13 class 𝑤
12		vv	. . . . . . . . . . . . . 14 setvar 𝑣
13	12	cv 1474	. . . . . . . . . . . . 13 class 𝑣
14	11, 13	cop 4131	. . . . . . . . . . . 12 class ⟨𝑤, 𝑣⟩
15		cer 9565	. . . . . . . . . . . 12 class ~R
16	14, 15	cec 7627	. . . . . . . . . . 11 class [⟨𝑤, 𝑣⟩] ~R
17	3, 16	wceq 1475	. . . . . . . . . 10 wff 𝑥 = [⟨𝑤, 𝑣⟩] ~R
18		vu	. . . . . . . . . . . . . 14 setvar 𝑢
19	18	cv 1474	. . . . . . . . . . . . 13 class 𝑢
20		vf	. . . . . . . . . . . . . 14 setvar 𝑓
21	20	cv 1474	. . . . . . . . . . . . 13 class 𝑓
22	19, 21	cop 4131	. . . . . . . . . . . 12 class ⟨𝑢, 𝑓⟩
23	22, 15	cec 7627	. . . . . . . . . . 11 class [⟨𝑢, 𝑓⟩] ~R
24	7, 23	wceq 1475	. . . . . . . . . 10 wff 𝑦 = [⟨𝑢, 𝑓⟩] ~R
25	17, 24	wa 383	. . . . . . . . 9 wff (𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~R )
26		vz	. . . . . . . . . . 11 setvar 𝑧
27	26	cv 1474	. . . . . . . . . 10 class 𝑧
28		cmp 9563	. . . . . . . . . . . . . 14 class ·P
29	11, 19, 28	co 6549	. . . . . . . . . . . . 13 class (𝑤 ·P 𝑢)
30	13, 21, 28	co 6549	. . . . . . . . . . . . 13 class (𝑣 ·P 𝑓)
31		cpp 9562	. . . . . . . . . . . . 13 class +P
32	29, 30, 31	co 6549	. . . . . . . . . . . 12 class ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓))
33	11, 21, 28	co 6549	. . . . . . . . . . . . 13 class (𝑤 ·P 𝑓)
34	13, 19, 28	co 6549	. . . . . . . . . . . . 13 class (𝑣 ·P 𝑢)
35	33, 34, 31	co 6549	. . . . . . . . . . . 12 class ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))
36	32, 35	cop 4131	. . . . . . . . . . 11 class ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩
37	36, 15	cec 7627	. . . . . . . . . 10 class [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R
38	27, 37	wceq 1475	. . . . . . . . 9 wff 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R
39	25, 38	wa 383	. . . . . . . 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 )
40	39, 20	wex 1695	. . . . . . 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 )
41	40, 18	wex 1695	. . . . . 6 wff ∃𝑢∃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 )
42	41, 12	wex 1695	. . . . 5 wff ∃𝑣∃𝑢∃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 )
43	42, 10	wex 1695	. . . 4 wff ∃𝑤∃𝑣∃𝑢∃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 )
44	9, 43	wa 383	. . 3 wff ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃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 ))
45	44, 2, 6, 26	coprab 6550	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃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 ))}
46	1, 45	wceq 1475	1 wff ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃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 ))}

This definition is referenced by:  mulsrpr  9776  dmmulsr  9786