Theorem mulidnq 7344

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 7303	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5858	. . 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 2185	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴))
5		df-1nqqs 7306	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
6	5	oveq2i 5862	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7		1pi 7270	. . . . 5 ⊢ 1o ∈ N
8		mulpipqqs 7328	. . . . 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 437	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
10	6, 9	eqtrid 2215	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
11		mulcompig 7286	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
12	7, 11	mpan 422	. . . . . 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 7286	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
15	7, 14	mpan 422	. . . . . 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 3771	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
18	17	eceq1d 6547	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19		mulcanenqec 7341	. . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
20	7, 19	mp3an1 1319	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
21	10, 18, 20	3eqtr2d 2209	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q )
22	1, 4, 21	ecoptocl 6598	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ⟨cop 3584  (class class class)co 5851  1_oc1o 6386  [cec 6509  Ncnpi 7227   ·_N cmi 7229   ~_Q ceq 7234  Qcnq 7235  1_Qc1q 7236   ·_Q cmq 7238

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-oadd 6397  df-omul 6398  df-er 6511  df-ec 6513  df-qs 6517  df-ni 7259  df-mi 7261  df-mpq 7300  df-enq 7302  df-nqqs 7303  df-mqqs 7305  df-1nqqs 7306

This theorem is referenced by:  recmulnqg  7346  rec1nq  7350  ltaddnq  7362  halfnqq  7365  prarloclemarch  7373  ltrnqg  7375  addnqprllem  7482  addnqprulem  7483  addnqprl  7484  addnqpru  7485  appdivnq  7518  prmuloc2  7522  mulnqprl  7523  mulnqpru  7524  1idprl  7545  1idpru  7546  recexprlem1ssl  7588  recexprlem1ssu  7589