Theorem addpipq 3848

Description: Addition of positive fractions in terms of positive integers.

Assertion

Ref	Expression
addpipq	⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → ([⟨A, B⟩] ~Q +Q [⟨C, D⟩] ~Q ) = [⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩] ~Q )

Proof of Theorem addpipq

Step	Hyp	Ref	Expression
1		opex 1893	. 2 ⊢ ⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩ ∈ V
2		opex 1893	. 2 ⊢ ⟨((a ·N h) +N (b ·N g)), (b ·N h)⟩ ∈ V
3		opex 1893	. 2 ⊢ ⟨((c ·N s) +N (d ·N t)), (d ·N s)⟩ ∈ V
4		enqex 3842	. 2 ⊢ ~Q ∈ V
5		enqer 3840	. 2 ⊢ Er ~Q
6		dmenq 3839	. 2 ⊢ dom ~Q = (N × N)
7		df-enq 3831	. 2 ⊢ ~Q = {⟨x, y⟩∣((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·N u) = (w ·N v)))}
8		opreq12 3008	. . . 4 ⊢ ((z = a ∧ u = d) → (z ·N u) = (a ·N d))
9		opreq12 3008	. . . 4 ⊢ ((w = b ∧ v = c) → (w ·N v) = (b ·N c))
10	8, 9	cleqan12d 1116	. . 3 ⊢ (((z = a ∧ u = d) ∧ (w = b ∧ v = c)) → ((z ·N u) = (w ·N v) ↔ (a ·N d) = (b ·N c)))
11	10	an42s 391	. 2 ⊢ (((z = a ∧ w = b) ∧ (v = c ∧ u = d)) → ((z ·N u) = (w ·N v) ↔ (a ·N d) = (b ·N c)))
12		opreq12 3008	. . . 4 ⊢ ((z = g ∧ u = s) → (z ·N u) = (g ·N s))
13		opreq12 3008	. . . 4 ⊢ ((w = h ∧ v = t) → (w ·N v) = (h ·N t))
14	12, 13	cleqan12d 1116	. . 3 ⊢ (((z = g ∧ u = s) ∧ (w = h ∧ v = t)) → ((z ·N u) = (w ·N v) ↔ (g ·N s) = (h ·N t)))
15	14	an42s 391	. 2 ⊢ (((z = g ∧ w = h) ∧ (v = t ∧ u = s)) → ((z ·N u) = (w ·N v) ↔ (g ·N s) = (h ·N t)))
16		df-plpq 3829	. 2 ⊢ +pQ = {⟨⟨x, y⟩, z⟩∣((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))}
17		opeq12 1878	. . 3 ⊢ ((((w ·N f) +N (v ·N u)) = ((a ·N h) +N (b ·N g)) ∧ (v ·N f) = (b ·N h)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((a ·N h) +N (b ·N g)), (b ·N h)⟩)
18		opreq12 3008	. . . . 5 ⊢ ((w = a ∧ f = h) → (w ·N f) = (a ·N h))
19		opreq12 3008	. . . . 5 ⊢ ((v = b ∧ u = g) → (v ·N u) = (b ·N g))
20	18, 19	opreqan12d 3015	. . . 4 ⊢ (((w = a ∧ f = h) ∧ (v = b ∧ u = g)) → ((w ·N f) +N (v ·N u)) = ((a ·N h) +N (b ·N g)))
21	20	an42s 391	. . 3 ⊢ (((w = a ∧ v = b) ∧ (u = g ∧ f = h)) → ((w ·N f) +N (v ·N u)) = ((a ·N h) +N (b ·N g)))
22		opreq12 3008	. . . . 5 ⊢ ((v = b ∧ f = h) → (v ·N f) = (b ·N h))
23	22	adantl 305	. . . 4 ⊢ (((w = a ∧ u = g) ∧ (v = b ∧ f = h)) → (v ·N f) = (b ·N h))
24	23	an4s 390	. . 3 ⊢ (((w = a ∧ v = b) ∧ (u = g ∧ f = h)) → (v ·N f) = (b ·N h))
25	17, 21, 24	sylanc 361	. 2 ⊢ (((w = a ∧ v = b) ∧ (u = g ∧ f = h)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((a ·N h) +N (b ·N g)), (b ·N h)⟩)
26		opeq12 1878	. . 3 ⊢ ((((w ·N f) +N (v ·N u)) = ((c ·N s) +N (d ·N t)) ∧ (v ·N f) = (d ·N s)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((c ·N s) +N (d ·N t)), (d ·N s)⟩)
27		opreq12 3008	. . . . 5 ⊢ ((w = c ∧ f = s) → (w ·N f) = (c ·N s))
28		opreq12 3008	. . . . 5 ⊢ ((v = d ∧ u = t) → (v ·N u) = (d ·N t))
29	27, 28	opreqan12d 3015	. . . 4 ⊢ (((w = c ∧ f = s) ∧ (v = d ∧ u = t)) → ((w ·N f) +N (v ·N u)) = ((c ·N s) +N (d ·N t)))
30	29	an42s 391	. . 3 ⊢ (((w = c ∧ v = d) ∧ (u = t ∧ f = s)) → ((w ·N f) +N (v ·N u)) = ((c ·N s) +N (d ·N t)))
31		opreq12 3008	. . . . 5 ⊢ ((v = d ∧ f = s) → (v ·N f) = (d ·N s))
32	31	adantl 305	. . . 4 ⊢ (((w = c ∧ u = t) ∧ (v = d ∧ f = s)) → (v ·N f) = (d ·N s))
33	32	an4s 390	. . 3 ⊢ (((w = c ∧ v = d) ∧ (u = t ∧ f = s)) → (v ·N f) = (d ·N s))
34	26, 30, 33	sylanc 361	. 2 ⊢ (((w = c ∧ v = d) ∧ (u = t ∧ f = s)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((c ·N s) +N (d ·N t)), (d ·N s)⟩)
35		opeq12 1878	. . 3 ⊢ ((((w ·N f) +N (v ·N u)) = ((A ·N D) +N (B ·N C)) ∧ (v ·N f) = (B ·N D)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩)
36		opreq12 3008	. . . . 5 ⊢ ((w = A ∧ f = D) → (w ·N f) = (A ·N D))
37		opreq12 3008	. . . . 5 ⊢ ((v = B ∧ u = C) → (v ·N u) = (B ·N C))
38	36, 37	opreqan12d 3015	. . . 4 ⊢ (((w = A ∧ f = D) ∧ (v = B ∧ u = C)) → ((w ·N f) +N (v ·N u)) = ((A ·N D) +N (B ·N C)))
39	38	an42s 391	. . 3 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → ((w ·N f) +N (v ·N u)) = ((A ·N D) +N (B ·N C)))
40		opreq12 3008	. . . . 5 ⊢ ((v = B ∧ f = D) → (v ·N f) = (B ·N D))
41	40	adantl 305	. . . 4 ⊢ (((w = A ∧ u = C) ∧ (v = B ∧ f = D)) → (v ·N f) = (B ·N D))
42	41	an4s 390	. . 3 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → (v ·N f) = (B ·N D))
43	35, 39, 42	sylanc 361	. 2 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩ = ⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩)
44		df-plq 3833	. 2 ⊢ +Q = {⟨⟨x, y⟩, z⟩∣((x ∈ Q ∧ y ∈ Q) ∧ ∃a∃b∃c∃d((x = [⟨a, b⟩] ~Q ∧ y = [⟨c, d⟩] ~Q ) ∧ z = [(⟨a, b⟩ +pQ ⟨c, d⟩)] ~Q ))}
45		df-nq 3832	. 2 ⊢ Q = ((N × N) / ~Q )
46		visset 1350	. . 3 ⊢ a ∈ V
47		visset 1350	. . 3 ⊢ b ∈ V
48		visset 1350	. . 3 ⊢ c ∈ V
49		visset 1350	. . 3 ⊢ d ∈ V
50		visset 1350	. . 3 ⊢ g ∈ V
51		visset 1350	. . 3 ⊢ h ∈ V
52		visset 1350	. . 3 ⊢ t ∈ V
53		visset 1350	. . 3 ⊢ s ∈ V
54	46, 47, 48, 49, 50, 51, 52, 53	addcmpblnq 3846	. 2 ⊢ ((((a ∈ N ∧ b ∈ N) ∧ (c ∈ N ∧ d ∈ N)) ∧ ((g ∈ N ∧ h ∈ N) ∧ (t ∈ N ∧ s ∈ N))) → (((a ·N d) = (b ·N c) ∧ (g ·N s) = (h ·N t)) → ⟨((a ·N h) +N (b ·N g)), (b ·N h)⟩ ~Q ⟨((c ·N s) +N (d ·N t)), (d ·N s)⟩))
55	1, 2, 3, 4, 5, 6, 7, 11, 15, 16, 25, 34, 43, 44, 45, 54	oprec 3254	1 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → ([⟨A, B⟩] ~Q +Q [⟨C, D⟩] ~Q ) = [⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩] ~Q )

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  [cec 3198  Ncnpi 3766   +_N cpli 3767   ·_N cmi 3768   +_pQ cplpq 3770   ~_Q ceq 3772  Qcnq 3773   +_Q cplq 3775

This theorem is referenced by:  addclpq 3852  addcompq 3856  addasspq 3857  distrpq 3861  ltapq 3870  1lt2pq 3872  ltexpq 3874  halfpq 3876  prlem934a 3931

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-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-plpq 3829  df-enq 3831  df-nq 3832  df-plq 3833

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