Theorem mulidpq 3863

Description: Multiplication identity element for positive fractions.

Assertion

Ref	Expression
mulidpq	⊢ (A ∈ Q → (A ·Q 1Q) = A)

Proof of Theorem mulidpq

Step	Hyp	Ref	Expression
1		df-nq 3832	. 2 ⊢ Q = ((N × N) / ~Q )
2		opreq1 3006	. . 3 ⊢ ([⟨x, y⟩] ~Q = A → ([⟨x, y⟩] ~Q ·Q 1Q) = (A ·Q 1Q))
3		id 9	. . 3 ⊢ ([⟨x, y⟩] ~Q = A → [⟨x, y⟩] ~Q = A)
4	2, 3	cleq12d 1115	. 2 ⊢ ([⟨x, y⟩] ~Q = A → (([⟨x, y⟩] ~Q ·Q 1Q) = [⟨x, y⟩] ~Q ↔ (A ·Q 1Q) = A))
5		1pi 3805	. . . . . 6 ⊢ 1o ∈ N
6	5, 5	pm3.2i 234	. . . . 5 ⊢ (1o ∈ N ∧ 1o ∈ N)
7		mulpipq 3849	. . . . 5 ⊢ (((x ∈ N ∧ y ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([⟨x, y⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(x ·N 1o), (y ·N 1o)⟩] ~Q )
8	6, 7	mpan2 519	. . . 4 ⊢ ((x ∈ N ∧ y ∈ N) → ([⟨x, y⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(x ·N 1o), (y ·N 1o)⟩] ~Q )
9		df-1q 3837	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
10	9	opreq2i 3010	. . . 4 ⊢ ([⟨x, y⟩] ~Q ·Q 1Q) = ([⟨x, y⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
11	5	elisseti 1355	. . . . . . 7 ⊢ 1o ∈ V
12		visset 1350	. . . . . . 7 ⊢ x ∈ V
13	11, 12	mulcompi 3818	. . . . . 6 ⊢ (1o ·N x) = (x ·N 1o)
14		visset 1350	. . . . . . 7 ⊢ y ∈ V
15	11, 14	mulcompi 3818	. . . . . 6 ⊢ (1o ·N y) = (y ·N 1o)
16		opeq12 1878	. . . . . 6 ⊢ (((1o ·N x) = (x ·N 1o) ∧ (1o ·N y) = (y ·N 1o)) → ⟨(1o ·N x), (1o ·N y)⟩ = ⟨(x ·N 1o), (y ·N 1o)⟩)
17	13, 15, 16	mp2an 520	. . . . 5 ⊢ ⟨(1o ·N x), (1o ·N y)⟩ = ⟨(x ·N 1o), (y ·N 1o)⟩
18		eceq2 3215	. . . . 5 ⊢ (⟨(1o ·N x), (1o ·N y)⟩ = ⟨(x ·N 1o), (y ·N 1o)⟩ → [⟨(1o ·N x), (1o ·N y)⟩] ~Q = [⟨(x ·N 1o), (y ·N 1o)⟩] ~Q )
19	17, 18	ax-mp 6	. . . 4 ⊢ [⟨(1o ·N x), (1o ·N y)⟩] ~Q = [⟨(x ·N 1o), (y ·N 1o)⟩] ~Q
20	8, 10, 19	3eqtr4g 1147	. . 3 ⊢ ((x ∈ N ∧ y ∈ N) → ([⟨x, y⟩] ~Q ·Q 1Q) = [⟨(1o ·N x), (1o ·N y)⟩] ~Q )
21	11, 12, 14	distrpqlem 3860	. . . 4 ⊢ ((1o ∈ N ∧ x ∈ N ∧ y ∈ N) → [⟨(1o ·N x), (1o ·N y)⟩] ~Q = [⟨x, y⟩] ~Q )
22	5, 21	mp3an1 639	. . 3 ⊢ ((x ∈ N ∧ y ∈ N) → [⟨(1o ·N x), (1o ·N y)⟩] ~Q = [⟨x, y⟩] ~Q )
23	20, 22	eqtrd 1128	. 2 ⊢ ((x ∈ N ∧ y ∈ N) → ([⟨x, y⟩] ~Q ·Q 1Q) = [⟨x, y⟩] ~Q )
24	1, 4, 23	ecoptocl 3239	1 ⊢ (A ∈ Q → (A ·Q 1Q) = A)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  1_oc1o 3099  [cec 3198  Ncnpi 3766   ·_N cmi 3768   ~_Q ceq 3772  Qcnq 3773  1_Qc1q 3774   ·_Q cmq 3776

This theorem is referenced by:  recmulpq 3864  ltaddpq 3873  1pr 3911  addclprlem1 3912  addclprlem2 3913  mulclprlem 3915  1idpr 3927  prlem934a 3931  prlem936a 3947  prlem936 3949  reclem3pr 3952

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-mi 3796  df-mpq 3830  df-enq 3831  df-nq 3832  df-mq 3834  df-1q 3837

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