Theorem addpipqqs 7525

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 7524	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
2		addpipqqslem 7524	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (𝑔 ∈ N ∧ ℎ ∈ N)) → ⟨((𝑎 ·N ℎ) +N (𝑏 ·N 𝑔)), (𝑏 ·N ℎ)⟩ ∈ (N × N))
3		addpipqqslem 7524	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4		enqex 7515	. 2 ⊢ ~Q ∈ V
5		enqer 7513	. 2 ⊢ ~Q Er (N × N)
6		df-enq 7502	. 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 5983	. . . 4 ⊢ ((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
8		oveq12 5983	. . . 4 ⊢ ((𝑤 = 𝑏 ∧ 𝑣 = 𝑐) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
9	7, 8	eqeqan12d 2225	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) ∧ (𝑤 = 𝑏 ∧ 𝑣 = 𝑐)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
10	9	an42s 591	. 2 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11		oveq12 5983	. . . 4 ⊢ ((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
12		oveq12 5983	. . . 4 ⊢ ((𝑤 = ℎ ∧ 𝑣 = 𝑡) → (𝑤 ·N 𝑣) = (ℎ ·N 𝑡))
13	11, 12	eqeqan12d 2225	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) ∧ (𝑤 = ℎ ∧ 𝑣 = 𝑡)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
14	13	an42s 591	. 2 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
15		dfplpq2 7509	. 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 5983	. . . . 5 ⊢ ((𝑤 = 𝑎 ∧ 𝑓 = ℎ) → (𝑤 ·N 𝑓) = (𝑎 ·N ℎ))
17		oveq12 5983	. . . . 5 ⊢ ((𝑣 = 𝑏 ∧ 𝑢 = 𝑔) → (𝑣 ·N 𝑢) = (𝑏 ·N 𝑔))
18	16, 17	oveqan12d 5993	. . . 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 591	. . 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 5983	. . . 4 ⊢ ((𝑣 = 𝑏 ∧ 𝑓 = ℎ) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
21	20	ad2ant2l 508	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
22	19, 21	opeq12d 3844	. 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 5983	. . . . 5 ⊢ ((𝑤 = 𝑐 ∧ 𝑓 = 𝑠) → (𝑤 ·N 𝑓) = (𝑐 ·N 𝑠))
24		oveq12 5983	. . . . 5 ⊢ ((𝑣 = 𝑑 ∧ 𝑢 = 𝑡) → (𝑣 ·N 𝑢) = (𝑑 ·N 𝑡))
25	23, 24	oveqan12d 5993	. . . 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 591	. . 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 5983	. . . 4 ⊢ ((𝑣 = 𝑑 ∧ 𝑓 = 𝑠) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
28	27	ad2ant2l 508	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
29	26, 28	opeq12d 3844	. 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 5983	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑓 = 𝐷) → (𝑤 ·N 𝑓) = (𝐴 ·N 𝐷))
31		oveq12 5983	. . . . 5 ⊢ ((𝑣 = 𝐵 ∧ 𝑢 = 𝐶) → (𝑣 ·N 𝑢) = (𝐵 ·N 𝐶))
32	30, 31	oveqan12d 5993	. . . 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 591	. . 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 5983	. . . 4 ⊢ ((𝑣 = 𝐵 ∧ 𝑓 = 𝐷) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
35	34	ad2ant2l 508	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
36	33, 35	opeq12d 3844	. 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 7504	. 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 7503	. 2 ⊢ Q = ((N × N) / ~Q )
39		addcmpblnq 7522	. 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 6758	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 104   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ⟨cop 3649  (class class class)co 5974  [cec 6648  Ncnpi 7427   +_N cpli 7428   ·_N cmi 7429   +_pQ cplpq 7431   ~_Q ceq 7434  Qcnq 7435   +_Q cplq 7437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-plpq 7499  df-enq 7502  df-nqqs 7503  df-plqqs 7504

This theorem is referenced by:  addclnq  7530  addcomnqg  7536  addassnqg  7537  distrnqg  7542  ltanqg  7555  1lt2nq  7561  ltexnqq  7563  nqnq0a  7609  addpinq1  7619