Theorem mulpipqqs 7687

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 7642	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
2		mulclpi 7642	. . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐷 ∈ N) → (𝐵 ·N 𝐷) ∈ N)
3		opelxpi 4780	. . . 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 592	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
6		mulclpi 7642	. . . 4 ⊢ ((𝑎 ∈ N ∧ 𝑔 ∈ N) → (𝑎 ·N 𝑔) ∈ N)
7		mulclpi 7642	. . . 4 ⊢ ((𝑏 ∈ N ∧ ℎ ∈ N) → (𝑏 ·N ℎ) ∈ N)
8		opelxpi 4780	. . . 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 592	. 2 ⊢ (((𝑎 ∈ N ∧ 𝑏 ∈ N) ∧ (𝑔 ∈ N ∧ ℎ ∈ N)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N ℎ)⟩ ∈ (N × N))
11		mulclpi 7642	. . . 4 ⊢ ((𝑐 ∈ N ∧ 𝑡 ∈ N) → (𝑐 ·N 𝑡) ∈ N)
12		mulclpi 7642	. . . 4 ⊢ ((𝑑 ∈ N ∧ 𝑠 ∈ N) → (𝑑 ·N 𝑠) ∈ N)
13		opelxpi 4780	. . . 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 592	. 2 ⊢ (((𝑐 ∈ N ∧ 𝑑 ∈ N) ∧ (𝑡 ∈ N ∧ 𝑠 ∈ N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
16		enqex 7674	. 2 ⊢ ~Q ∈ V
17		enqer 7672	. 2 ⊢ ~Q Er (N × N)
18		df-enq 7661	. 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 533	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑢 = 𝑑)
21	19, 20	oveq12d 6067	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
22		simplr 529	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑤 = 𝑏)
23		simprl 531	. . . 4 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → 𝑣 = 𝑐)
24	22, 23	oveq12d 6067	. . 3 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
25	21, 24	eqeq12d 2247	. 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 533	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑢 = 𝑠)
28	26, 27	oveq12d 6067	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
29		simplr 529	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑤 = ℎ)
30		simprl 531	. . . 4 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → 𝑣 = 𝑡)
31	29, 30	oveq12d 6067	. . 3 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝑤 ·N 𝑣) = (ℎ ·N 𝑡))
32	28, 31	eqeq12d 2247	. 2 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = (ℎ ·N 𝑡)))
33		dfmpq2 7669	. 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 531	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑢 = 𝑔)
36	34, 35	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑤 ·N 𝑢) = (𝑎 ·N 𝑔))
37		simplr 529	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑣 = 𝑏)
38		simprr 533	. . . 4 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝑓 = ℎ)
39	37, 38	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → (𝑣 ·N 𝑓) = (𝑏 ·N ℎ))
40	36, 39	opeq12d 3890	. 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 531	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑢 = 𝑡)
43	41, 42	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑤 ·N 𝑢) = (𝑐 ·N 𝑡))
44		simplr 529	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑣 = 𝑑)
45		simprr 533	. . . 4 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝑓 = 𝑠)
46	44, 45	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
47	43, 46	opeq12d 3890	. 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 531	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑢 = 𝐶)
50	48, 49	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑤 ·N 𝑢) = (𝐴 ·N 𝐶))
51		simplr 529	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑣 = 𝐵)
52		simprr 533	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑓 = 𝐷)
53	51, 52	oveq12d 6067	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
54	50, 53	opeq12d 3890	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
55		df-mqqs 7664	. 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 7662	. 2 ⊢ Q = ((N × N) / ~Q )
57		mulcmpblnq 7682	. 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 6874	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 1398   ∈ wcel 2203  ⟨cop 3691   × cxp 4746  (class class class)co 6049  [cec 6764  Ncnpi 7586   ·_N cmi 7588   ·_pQ cmpq 7591   ~_Q ceq 7593  Qcnq 7594   ·_Q cmq 7597

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-mi 7620  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-mqqs 7664

This theorem is referenced by:  mulclnq  7690  mulcomnqg  7697  mulassnqg  7698  distrnqg  7701  mulidnq  7703  recexnq  7704  ltmnqg  7715  nqnq0m  7769