Theorem addpipqqs 7202

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 7201	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
2		addpipqqslem 7201	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (𝑔 ∈ N ∧ ℎ ∈ N)) → ⟨((𝑎 ·N ℎ) +N (𝑏 ·N 𝑔)), (𝑏 ·N ℎ)⟩ ∈ (N × N))
3		addpipqqslem 7201	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4		enqex 7192	. 2 ⊢ ~Q ∈ V
5		enqer 7190	. 2 ⊢ ~Q Er (N × N)
6		df-enq 7179	. 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 5791	. . . 4 ⊢ ((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
8		oveq12 5791	. . . 4 ⊢ ((𝑤 = 𝑏 ∧ 𝑣 = 𝑐) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
9	7, 8	eqeqan12d 2156	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) ∧ (𝑤 = 𝑏 ∧ 𝑣 = 𝑐)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
10	9	an42s 579	. 2 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11		oveq12 5791	. . . 4 ⊢ ((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
12		oveq12 5791	. . . 4 ⊢ ((𝑤 = ℎ ∧ 𝑣 = 𝑡) → (𝑤 ·N 𝑣) = (ℎ ·N 𝑡))
13	11, 12	eqeqan12d 2156	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) ∧ (𝑤 = ℎ ∧ 𝑣 = 𝑡)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
14	13	an42s 579	. 2 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
15		dfplpq2 7186	. 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 5791	. . . . 5 ⊢ ((𝑤 = 𝑎 ∧ 𝑓 = ℎ) → (𝑤 ·N 𝑓) = (𝑎 ·N ℎ))
17		oveq12 5791	. . . . 5 ⊢ ((𝑣 = 𝑏 ∧ 𝑢 = 𝑔) → (𝑣 ·N 𝑢) = (𝑏 ·N 𝑔))
18	16, 17	oveqan12d 5801	. . . 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 579	. . 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 5791	. . . 4 ⊢ ((𝑣 = 𝑏 ∧ 𝑓 = ℎ) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
21	20	ad2ant2l 500	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
22	19, 21	opeq12d 3721	. 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 5791	. . . . 5 ⊢ ((𝑤 = 𝑐 ∧ 𝑓 = 𝑠) → (𝑤 ·N 𝑓) = (𝑐 ·N 𝑠))
24		oveq12 5791	. . . . 5 ⊢ ((𝑣 = 𝑑 ∧ 𝑢 = 𝑡) → (𝑣 ·N 𝑢) = (𝑑 ·N 𝑡))
25	23, 24	oveqan12d 5801	. . . 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 579	. . 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 5791	. . . 4 ⊢ ((𝑣 = 𝑑 ∧ 𝑓 = 𝑠) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
28	27	ad2ant2l 500	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
29	26, 28	opeq12d 3721	. 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 5791	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑓 = 𝐷) → (𝑤 ·N 𝑓) = (𝐴 ·N 𝐷))
31		oveq12 5791	. . . . 5 ⊢ ((𝑣 = 𝐵 ∧ 𝑢 = 𝐶) → (𝑣 ·N 𝑢) = (𝐵 ·N 𝐶))
32	30, 31	oveqan12d 5801	. . . 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 579	. . 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 5791	. . . 4 ⊢ ((𝑣 = 𝐵 ∧ 𝑓 = 𝐷) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
35	34	ad2ant2l 500	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
36	33, 35	opeq12d 3721	. 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 7181	. 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 7180	. 2 ⊢ Q = ((N × N) / ~Q )
39		addcmpblnq 7199	. 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 6543	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 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ⟨cop 3535  (class class class)co 5782  [cec 6435  Ncnpi 7104   +_N cpli 7105   ·_N cmi 7106   +_pQ cplpq 7108   ~_Q ceq 7111  Qcnq 7112   +_Q cplq 7114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-plpq 7176  df-enq 7179  df-nqqs 7180  df-plqqs 7181

This theorem is referenced by:  addclnq  7207  addcomnqg  7213  addassnqg  7214  distrnqg  7219  ltanqg  7232  1lt2nq  7238  ltexnqq  7240  nqnq0a  7286  addpinq1  7296