Theorem addpipqqs 6560

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 6559	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
2		addpipqqslem 6559	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (𝑔 ∈ N ∧ ℎ ∈ N)) → ⟨((𝑎 ·N ℎ) +N (𝑏 ·N 𝑔)), (𝑏 ·N ℎ)⟩ ∈ (N × N))
3		addpipqqslem 6559	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4		enqex 6550	. 2 ⊢ ~Q ∈ V
5		enqer 6548	. 2 ⊢ ~Q Er (N × N)
6		df-enq 6537	. 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 5541	. . . 4 ⊢ ((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
8		oveq12 5541	. . . 4 ⊢ ((𝑤 = 𝑏 ∧ 𝑣 = 𝑐) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
9	7, 8	eqeqan12d 2096	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑢 = 𝑑) ∧ (𝑤 = 𝑏 ∧ 𝑣 = 𝑐)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
10	9	an42s 553	. 2 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11		oveq12 5541	. . . 4 ⊢ ((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
12		oveq12 5541	. . . 4 ⊢ ((𝑤 = ℎ ∧ 𝑣 = 𝑡) → (𝑤 ·N 𝑣) = (ℎ ·N 𝑡))
13	11, 12	eqeqan12d 2096	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑢 = 𝑠) ∧ (𝑤 = ℎ ∧ 𝑣 = 𝑡)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
14	13	an42s 553	. 2 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
15		dfplpq2 6544	. 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 5541	. . . . 5 ⊢ ((𝑤 = 𝑎 ∧ 𝑓 = ℎ) → (𝑤 ·N 𝑓) = (𝑎 ·N ℎ))
17		oveq12 5541	. . . . 5 ⊢ ((𝑣 = 𝑏 ∧ 𝑢 = 𝑔) → (𝑣 ·N 𝑢) = (𝑏 ·N 𝑔))
18	16, 17	oveqan12d 5551	. . . 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 553	. . 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 5541	. . . 4 ⊢ ((𝑣 = 𝑏 ∧ 𝑓 = ℎ) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
21	20	ad2ant2l 491	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
22	19, 21	opeq12d 3578	. 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 5541	. . . . 5 ⊢ ((𝑤 = 𝑐 ∧ 𝑓 = 𝑠) → (𝑤 ·N 𝑓) = (𝑐 ·N 𝑠))
24		oveq12 5541	. . . . 5 ⊢ ((𝑣 = 𝑑 ∧ 𝑢 = 𝑡) → (𝑣 ·N 𝑢) = (𝑑 ·N 𝑡))
25	23, 24	oveqan12d 5551	. . . 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 553	. . 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 5541	. . . 4 ⊢ ((𝑣 = 𝑑 ∧ 𝑓 = 𝑠) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
28	27	ad2ant2l 491	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
29	26, 28	opeq12d 3578	. 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 5541	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑓 = 𝐷) → (𝑤 ·N 𝑓) = (𝐴 ·N 𝐷))
31		oveq12 5541	. . . . 5 ⊢ ((𝑣 = 𝐵 ∧ 𝑢 = 𝐶) → (𝑣 ·N 𝑢) = (𝐵 ·N 𝐶))
32	30, 31	oveqan12d 5551	. . . 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 553	. . 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 5541	. . . 4 ⊢ ((𝑣 = 𝐵 ∧ 𝑓 = 𝐷) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
35	34	ad2ant2l 491	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
36	33, 35	opeq12d 3578	. 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 6539	. 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 6538	. 2 ⊢ Q = ((N × N) / ~Q )
39		addcmpblnq 6557	. 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 6235	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 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ⟨cop 3401  (class class class)co 5532  [cec 6127  Ncnpi 6462   +_N cpli 6463   ·_N cmi 6464   +_pQ cplpq 6466   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-plpq 6534  df-enq 6537  df-nqqs 6538  df-plqqs 6539

This theorem is referenced by:  addclnq  6565  addcomnqg  6571  addassnqg  6572  distrnqg  6577  ltanqg  6590  1lt2nq  6596  ltexnqq  6598  nqnq0a  6644  addpinq1  6654