Theorem mulidnq 7351

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 7310	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5860	. . 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 7313	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
6	5	oveq2i 5864	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7		1pi 7277	. . . . 5 ⊢ 1o ∈ N
8		mulpipqqs 7335	. . . . 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 7293	. . . . . . 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 7293	. . . . . . 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 3773	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
18	17	eceq1d 6549	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19		mulcanenqec 7348	. . . 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 6600	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ⟨cop 3586  (class class class)co 5853  1_oc1o 6388  [cec 6511  Ncnpi 7234   ·_N cmi 7236   ~_Q ceq 7241  Qcnq 7242  1_Qc1q 7243   ·_Q cmq 7245

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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313

This theorem is referenced by:  recmulnqg  7353  rec1nq  7357  ltaddnq  7369  halfnqq  7372  prarloclemarch  7380  ltrnqg  7382  addnqprllem  7489  addnqprulem  7490  addnqprl  7491  addnqpru  7492  appdivnq  7525  prmuloc2  7529  mulnqprl  7530  mulnqpru  7531  1idprl  7552  1idpru  7553  recexprlem1ssl  7595  recexprlem1ssu  7596