Theorem mulidnq 7190

Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)

Assertion

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

Proof of Theorem mulidnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nqqs 7149	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5774	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = (𝐴 ·Q 1Q))
3		id 19	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → [⟨𝑥, 𝑦⟩] ~Q = 𝐴)
4	2, 3	eqeq12d 2152	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴))
5		df-1nqqs 7152	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
6	5	oveq2i 5778	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7		1pi 7116	. . . . 5 ⊢ 1o ∈ N
8		mulpipqqs 7174	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
9	7, 7, 8	mpanr12 435	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
10	6, 9	syl5eq 2182	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
11		mulcompig 7132	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
12	7, 11	mpan 420	. . . . . 6 ⊢ (𝑥 ∈ N → (1o ·N 𝑥) = (𝑥 ·N 1o))
13	12	adantr 274	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
14		mulcompig 7132	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
15	7, 14	mpan 420	. . . . . 6 ⊢ (𝑦 ∈ N → (1o ·N 𝑦) = (𝑦 ·N 1o))
16	15	adantl 275	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
17	13, 16	opeq12d 3708	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
18	17	eceq1d 6458	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19		mulcanenqec 7187	. . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
20	7, 19	mp3an1 1302	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
21	10, 18, 20	3eqtr2d 2176	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q )
22	1, 4, 21	ecoptocl 6509	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ⟨cop 3525  (class class class)co 5767  1_oc1o 6299  [cec 6420  Ncnpi 7073   ·_N cmi 7075   ~_Q ceq 7080  Qcnq 7081  1_Qc1q 7082   ·_Q cmq 7084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152

This theorem is referenced by:  recmulnqg  7192  rec1nq  7196  ltaddnq  7208  halfnqq  7211  prarloclemarch  7219  ltrnqg  7221  addnqprllem  7328  addnqprulem  7329  addnqprl  7330  addnqpru  7331  appdivnq  7364  prmuloc2  7368  mulnqprl  7369  mulnqpru  7370  1idprl  7391  1idpru  7392  recexprlem1ssl  7434  recexprlem1ssu  7435