Theorem addpipqqs 6354

Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)

Assertion

Ref	Expression
addpipqqs	⊢ (((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 addpipqqs

Dummy variables x y z w v u 𝑡 𝑠 f g ℎ 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addpipqqslem 6353	. 2 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)⟩ ∈ (N × N))
2		addpipqqslem 6353	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (g ∈ N ∧ ℎ ∈ N)) → ⟨((𝑎 ·N ℎ) +N (𝑏 ·N g)), (𝑏 ·N ℎ)⟩ ∈ (N × N))
3		addpipqqslem 6353	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4		enqex 6344	. 2 ⊢ ~Q ∈ V
5		enqer 6342	. 2 ⊢ ~Q Er (N × N)
6		df-enq 6331	. 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)))}
7		oveq12 5464	. . . 4 ⊢ ((z = 𝑎 ∧ u = 𝑑) → (z ·N u) = (𝑎 ·N 𝑑))
8		oveq12 5464	. . . 4 ⊢ ((w = 𝑏 ∧ v = 𝑐) → (w ·N v) = (𝑏 ·N 𝑐))
9	7, 8	eqeqan12d 2052	. . 3 ⊢ (((z = 𝑎 ∧ u = 𝑑) ∧ (w = 𝑏 ∧ v = 𝑐)) → ((z ·N u) = (w ·N v) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
10	9	an42s 523	. 2 ⊢ (((z = 𝑎 ∧ w = 𝑏) ∧ (v = 𝑐 ∧ u = 𝑑)) → ((z ·N u) = (w ·N v) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11		oveq12 5464	. . . 4 ⊢ ((z = g ∧ u = 𝑠) → (z ·N u) = (g ·N 𝑠))
12		oveq12 5464	. . . 4 ⊢ ((w = ℎ ∧ v = 𝑡) → (w ·N v) = (ℎ ·N 𝑡))
13	11, 12	eqeqan12d 2052	. . 3 ⊢ (((z = g ∧ u = 𝑠) ∧ (w = ℎ ∧ v = 𝑡)) → ((z ·N u) = (w ·N v) ↔ (g ·N 𝑠) = (ℎ ·N 𝑡)))
14	13	an42s 523	. 2 ⊢ (((z = g ∧ w = ℎ) ∧ (v = 𝑡 ∧ u = 𝑠)) → ((z ·N u) = (w ·N v) ↔ (g ·N 𝑠) = (ℎ ·N 𝑡)))
15		dfplpq2 6338	. 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)⟩))}
16		oveq12 5464	. . . . 5 ⊢ ((w = 𝑎 ∧ f = ℎ) → (w ·N f) = (𝑎 ·N ℎ))
17		oveq12 5464	. . . . 5 ⊢ ((v = 𝑏 ∧ u = g) → (v ·N u) = (𝑏 ·N g))
18	16, 17	oveqan12d 5474	. . . 4 ⊢ (((w = 𝑎 ∧ f = ℎ) ∧ (v = 𝑏 ∧ u = g)) → ((w ·N f) +N (v ·N u)) = ((𝑎 ·N ℎ) +N (𝑏 ·N g)))
19	18	an42s 523	. . 3 ⊢ (((w = 𝑎 ∧ v = 𝑏) ∧ (u = g ∧ f = ℎ)) → ((w ·N f) +N (v ·N u)) = ((𝑎 ·N ℎ) +N (𝑏 ·N g)))
20		oveq12 5464	. . . 4 ⊢ ((v = 𝑏 ∧ f = ℎ) → (v ·N f) = (𝑏 ·N ℎ))
21	20	ad2ant2l 477	. . 3 ⊢ (((w = 𝑎 ∧ v = 𝑏) ∧ (u = g ∧ f = ℎ)) → (v ·N f) = (𝑏 ·N ℎ))
22	19, 21	opeq12d 3548	. 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)⟩)
23		oveq12 5464	. . . . 5 ⊢ ((w = 𝑐 ∧ f = 𝑠) → (w ·N f) = (𝑐 ·N 𝑠))
24		oveq12 5464	. . . . 5 ⊢ ((v = 𝑑 ∧ u = 𝑡) → (v ·N u) = (𝑑 ·N 𝑡))
25	23, 24	oveqan12d 5474	. . . 4 ⊢ (((w = 𝑐 ∧ f = 𝑠) ∧ (v = 𝑑 ∧ u = 𝑡)) → ((w ·N f) +N (v ·N u)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
26	25	an42s 523	. . 3 ⊢ (((w = 𝑐 ∧ v = 𝑑) ∧ (u = 𝑡 ∧ f = 𝑠)) → ((w ·N f) +N (v ·N u)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
27		oveq12 5464	. . . 4 ⊢ ((v = 𝑑 ∧ f = 𝑠) → (v ·N f) = (𝑑 ·N 𝑠))
28	27	ad2ant2l 477	. . 3 ⊢ (((w = 𝑐 ∧ v = 𝑑) ∧ (u = 𝑡 ∧ f = 𝑠)) → (v ·N f) = (𝑑 ·N 𝑠))
29	26, 28	opeq12d 3548	. 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)⟩)
30		oveq12 5464	. . . . 5 ⊢ ((w = A ∧ f = 𝐷) → (w ·N f) = (A ·N 𝐷))
31		oveq12 5464	. . . . 5 ⊢ ((v = B ∧ u = 𝐶) → (v ·N u) = (B ·N 𝐶))
32	30, 31	oveqan12d 5474	. . . 4 ⊢ (((w = A ∧ f = 𝐷) ∧ (v = B ∧ u = 𝐶)) → ((w ·N f) +N (v ·N u)) = ((A ·N 𝐷) +N (B ·N 𝐶)))
33	32	an42s 523	. . 3 ⊢ (((w = A ∧ v = B) ∧ (u = 𝐶 ∧ f = 𝐷)) → ((w ·N f) +N (v ·N u)) = ((A ·N 𝐷) +N (B ·N 𝐶)))
34		oveq12 5464	. . . 4 ⊢ ((v = B ∧ f = 𝐷) → (v ·N f) = (B ·N 𝐷))
35	34	ad2ant2l 477	. . 3 ⊢ (((w = A ∧ v = B) ∧ (u = 𝐶 ∧ f = 𝐷)) → (v ·N f) = (B ·N 𝐷))
36	33, 35	opeq12d 3548	. 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)⟩)
37		df-plqqs 6333	. 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 ))}
38		df-nqqs 6332	. 2 ⊢ Q = ((N × N) / ~Q )
39		addcmpblnq 6351	. 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)⟩))
40	1, 2, 3, 4, 5, 6, 10, 14, 15, 22, 29, 36, 37, 38, 39	oviec 6148	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 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  (class class class)co 5455  [cec 6040  Ncnpi 6256   +_N cpli 6257   ·_N cmi 6258   +_pQ cplpq 6260   ~_Q ceq 6263  Qcnq 6264   +_Q cplq 6266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-plpq 6328  df-enq 6331  df-nqqs 6332  df-plqqs 6333

This theorem is referenced by:  addclnq  6359  addcomnqg  6365  addassnqg  6366  distrnqg  6371  ltanqg  6384  1lt2nq  6389  ltexnqq  6391  nqnq0a  6436  addpinq1  6446