Theorem mulsrpr 3979

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

Assertion

Ref	Expression
mulsrpr	⊢ (((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ 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))⟩ ∈ V
2		opex 1893	. 2 ⊢ ⟨((a ·P g) +P (b ·P h)), ((a ·P h) +P (b ·P g))⟩ ∈ V
3		opex 1893	. 2 ⊢ ⟨((c ·P t) +P (d ·P s)), ((c ·P s) +P (d ·P t))⟩ ∈ V
4		enrex 3972	. 2 ⊢ ~R ∈ V
5		enrer 3970	. 2 ⊢ Er ~R
6		dmenr 3969	. 2 ⊢ dom ~R = (P × P)
7		df-enr 3960	. 2 ⊢ ~R = {⟨x, y⟩∣((x ∈ (P × P) ∧ y ∈ (P × P)) ∧ ∃z∃w∃v∃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 ∈ (P × P) ∧ y ∈ (P × P)) ∧ ∃w∃v∃u∃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))⟩ = ⟨((A ·P C) +P (B ·P D)), ((A ·P D) +P (B ·P C))⟩)
47		df-mr 3963	. 2 ⊢ ·R = {⟨⟨x, y⟩, z⟩∣((x ∈ R ∧ y ∈ R) ∧ ∃a∃b∃c∃d((x = [⟨a, b⟩] ~R ∧ y = [⟨c, d⟩] ~R ) ∧ z = [(⟨a, b⟩ ·pR ⟨c, d⟩)] ~R ))}
48		df-nr 3961	. 2 ⊢ R = ((P × P) / ~R )
49		visset 1350	. . 3 ⊢ a ∈ V
50		visset 1350	. . 3 ⊢ b ∈ V
51		visset 1350	. . 3 ⊢ c ∈ V
52		visset 1350	. . 3 ⊢ d ∈ V
53		visset 1350	. . 3 ⊢ g ∈ V
54		visset 1350	. . 3 ⊢ h ∈ V
55		visset 1350	. . 3 ⊢ t ∈ V
56		visset 1350	. . 3 ⊢ s ∈ V
57	49, 50, 51, 52, 53, 54, 55, 56	mulcmpblnr 3977	. 2 ⊢ ((((a ∈ P ∧ b ∈ P) ∧ (c ∈ P ∧ d ∈ P)) ∧ ((g ∈ P ∧ h ∈ P) ∧ (t ∈ P ∧ s ∈ 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))⟩))
58	1, 2, 3, 4, 5, 6, 7, 11, 15, 16, 26, 36, 46, 47, 48, 57	oprec 3254	1 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ 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 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  [cec 3198  Pcnp 3779   +_P cpp 3781   ·_P cmp 3782   ·_pR cmpr 3785   ~_R cer 3786  Rcnr 3787   ·_R cmr 3792

This theorem is referenced by:  mulclsr 3987  mulcomsr 3992  mulasssr 3993  distrsr 3994  m1m1sr 3996  1idsr 4001  00sr 4002  recexsrlem 4006  mulgt0sr 4008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-mp 3883  df-ltp 3884  df-mpr 3959  df-enr 3960  df-nr 3961  df-mr 3963

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