Theorem dmmulpq 6364

Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
dmmulpq	⊢ dom ·Q = (Q × Q)

Proof of Theorem dmmulpq

Dummy variables x y z v w u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmoprab 5527	. . 3 ⊢ dom {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))} = {⟨x, y⟩ ∣ ∃z((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))}
2		df-mqqs 6334	. . . 4 ⊢ ·Q = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))}
3	2	dmeqi 4479	. . 3 ⊢ dom ·Q = dom {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))}
4		dmaddpqlem 6361	. . . . . . . . 9 ⊢ (x ∈ Q → ∃w∃v x = [⟨w, v⟩] ~Q )
5		dmaddpqlem 6361	. . . . . . . . 9 ⊢ (y ∈ Q → ∃u∃f y = [⟨u, f⟩] ~Q )
6	4, 5	anim12i 321	. . . . . . . 8 ⊢ ((x ∈ Q ∧ y ∈ Q) → (∃w∃v x = [⟨w, v⟩] ~Q ∧ ∃u∃f y = [⟨u, f⟩] ~Q ))
7		ee4anv 1806	. . . . . . . 8 ⊢ (∃w∃v∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ↔ (∃w∃v x = [⟨w, v⟩] ~Q ∧ ∃u∃f y = [⟨u, f⟩] ~Q ))
8	6, 7	sylibr 137	. . . . . . 7 ⊢ ((x ∈ Q ∧ y ∈ Q) → ∃w∃v∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ))
9		enqex 6344	. . . . . . . . . . . . . 14 ⊢ ~Q ∈ V
10		ecexg 6046	. . . . . . . . . . . . . 14 ⊢ ( ~Q ∈ V → [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ∈ V)
11	9, 10	ax-mp 7	. . . . . . . . . . . . 13 ⊢ [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ∈ V
12	11	isseti 2557	. . . . . . . . . . . 12 ⊢ ∃z z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q
13		ax-ia3 101	. . . . . . . . . . . . 13 ⊢ ((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → (z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q → ((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )))
14	13	eximdv 1757	. . . . . . . . . . . 12 ⊢ ((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → (∃z z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q → ∃z((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )))
15	12, 14	mpi 15	. . . . . . . . . . 11 ⊢ ((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → ∃z((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
16	15	2eximi 1489	. . . . . . . . . 10 ⊢ (∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → ∃u∃f∃z((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
17		exrot3 1577	. . . . . . . . . 10 ⊢ (∃z∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ) ↔ ∃u∃f∃z((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
18	16, 17	sylibr 137	. . . . . . . . 9 ⊢ (∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → ∃z∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
19	18	2eximi 1489	. . . . . . . 8 ⊢ (∃w∃v∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → ∃w∃v∃z∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
20		exrot3 1577	. . . . . . . 8 ⊢ (∃z∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ) ↔ ∃w∃v∃z∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
21	19, 20	sylibr 137	. . . . . . 7 ⊢ (∃w∃v∃u∃f(x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) → ∃z∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
22	8, 21	syl 14	. . . . . 6 ⊢ ((x ∈ Q ∧ y ∈ Q) → ∃z∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))
23	22	pm4.71i 371	. . . . 5 ⊢ ((x ∈ Q ∧ y ∈ Q) ↔ ((x ∈ Q ∧ y ∈ Q) ∧ ∃z∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )))
24		19.42v 1783	. . . . 5 ⊢ (∃z((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )) ↔ ((x ∈ Q ∧ y ∈ Q) ∧ ∃z∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )))
25	23, 24	bitr4i 176	. . . 4 ⊢ ((x ∈ Q ∧ y ∈ Q) ↔ ∃z((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q )))
26	25	opabbii 3815	. . 3 ⊢ {⟨x, y⟩ ∣ (x ∈ Q ∧ y ∈ Q)} = {⟨x, y⟩ ∣ ∃z((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))}
27	1, 3, 26	3eqtr4i 2067	. 2 ⊢ dom ·Q = {⟨x, y⟩ ∣ (x ∈ Q ∧ y ∈ Q)}
28		df-xp 4294	. 2 ⊢ (Q × Q) = {⟨x, y⟩ ∣ (x ∈ Q ∧ y ∈ Q)}
29	27, 28	eqtr4i 2060	1 ⊢ dom ·Q = (Q × Q)

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370  {copab 3808   × cxp 4286  dom cdm 4288  (class class class)co 5455  {coprab 5456  [cec 6040   ·_pQ cmpq 6261   ~_Q ceq 6263  Qcnq 6264   ·_Q cmq 6267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-iom 4257  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-oprab 5459  df-ec 6044  df-qs 6048  df-ni 6288  df-enq 6331  df-nqqs 6332  df-mqqs 6334

This theorem is referenced by: (None)