Theorem mulpipqqs 7457

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

Assertion

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

Proof of Theorem mulpipqqs

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		mulclpi 7412	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
2		mulclpi 7412	. . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐷 ∈ N) → (𝐵 ·N 𝐷) ∈ N)
3		opelxpi 4696	. . . 4 ⊢ (((𝐴 ·N 𝐶) ∈ N ∧ (𝐵 ·N 𝐷) ∈ N) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
4	1, 2, 3	syl2an 289	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐶 ∈ N) ∧ (𝐵 ∈ N ∧ 𝐷 ∈ N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
5	4	an4s 588	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
6		mulclpi 7412	. . . 4 ⊢ ((𝑎 ∈ N ∧ 𝑔 ∈ N) → (𝑎 ·N 𝑔) ∈ N)
7		mulclpi 7412	. . . 4 ⊢ ((𝑏 ∈ N ∧ ℎ ∈ N) → (𝑏 ·N ℎ) ∈ N)
8		opelxpi 4696	. . . 4 ⊢ (((𝑎 ·N 𝑔) ∈ N ∧ (𝑏 ·N ℎ) ∈ N) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N ℎ)⟩ ∈ (N × N))
9	6, 7, 8	syl2an 289	. . 3 ⊢ (((𝑎 ∈ N ∧ 𝑔 ∈ N) ∧ (𝑏 ∈ N ∧ ℎ ∈ N)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N ℎ)⟩ ∈ (N × N))
10	9	an4s 588	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (𝑔 ∈ N ∧ ℎ ∈ N)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N ℎ)⟩ ∈ (N × N))
11		mulclpi 7412	. . . 4 ⊢ ((𝑐 ∈ N ∧ 𝑡 ∈ N) → (𝑐 ·N 𝑡) ∈ N)
12		mulclpi 7412	. . . 4 ⊢ ((𝑑 ∈ N ∧ 𝑠 ∈ N) → (𝑑 ·N 𝑠) ∈ N)
13		opelxpi 4696	. . . 4 ⊢ (((𝑐 ·N 𝑡) ∈ N ∧ (𝑑 ·N 𝑠) ∈ N) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
14	11, 12, 13	syl2an 289	. . 3 ⊢ (((𝑐 ∈ N ∧ 𝑡 ∈ N) ∧ (𝑑 ∈ N ∧ 𝑠 ∈ N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
15	14	an4s 588	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
16		enqex 7444	. 2 ⊢ ~Q ∈ V
17		enqer 7442	. 2 ⊢ ~Q Er (N × N)
18		df-enq 7431	. 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)))}
19		simpll 527	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑧 = 𝑎)
20		simprr 531	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑢 = 𝑑)
21	19, 20	oveq12d 5943	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
22		simplr 528	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑤 = 𝑏)
23		simprl 529	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑣 = 𝑐)
24	22, 23	oveq12d 5943	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
25	21, 24	eqeq12d 2211	. 2 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
26		simpll 527	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑧 = 𝑔)
27		simprr 531	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑢 = 𝑠)
28	26, 27	oveq12d 5943	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
29		simplr 528	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑤 = ℎ)
30		simprl 529	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑣 = 𝑡)
31	29, 30	oveq12d 5943	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝑤 ·N 𝑣) = (ℎ ·N 𝑡))
32	28, 31	eqeq12d 2211	. 2 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
33		dfmpq2 7439	. 2 ⊢ ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}
34		simpll 527	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑤 = 𝑎)
35		simprl 529	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑢 = 𝑔)
36	34, 35	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑤 ·N 𝑢) = (𝑎 ·N 𝑔))
37		simplr 528	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑣 = 𝑏)
38		simprr 531	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑓 = ℎ)
39	37, 38	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
40	36, 39	opeq12d 3817	. 2 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑎 ·N 𝑔), (𝑏 ·N ℎ)⟩)
41		simpll 527	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑤 = 𝑐)
42		simprl 529	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑢 = 𝑡)
43	41, 42	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑤 ·N 𝑢) = (𝑐 ·N 𝑡))
44		simplr 528	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑣 = 𝑑)
45		simprr 531	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑓 = 𝑠)
46	44, 45	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
47	43, 46	opeq12d 3817	. 2 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩)
48		simpll 527	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑤 = 𝐴)
49		simprl 529	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑢 = 𝐶)
50	48, 49	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑤 ·N 𝑢) = (𝐴 ·N 𝐶))
51		simplr 528	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑣 = 𝐵)
52		simprr 531	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑓 = 𝐷)
53	51, 52	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
54	50, 53	opeq12d 3817	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
55		df-mqqs 7434	. 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 ))}
56		df-nqqs 7432	. 2 ⊢ Q = ((N × N) / ~Q )
57		mulcmpblnq 7452	. 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 𝑔), (𝑏 ·N ℎ)⟩ ~Q ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩))
58	5, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57	oviec 6709	1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ⟨cop 3626   × cxp 4662  (class class class)co 5925  [cec 6599  Ncnpi 7356   ·_N cmi 7358   ·_pQ cmpq 7361   ~_Q ceq 7363  Qcnq 7364   ·_Q cmq 7367

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-mi 7390  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-mqqs 7434

This theorem is referenced by:  mulclnq  7460  mulcomnqg  7467  mulassnqg  7468  distrnqg  7471  mulidnq  7473  recexnq  7474  ltmnqg  7485  nqnq0m  7539