Theorem mulidnq 7609

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 7568	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 6025	. . 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 2246	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴))
5		df-1nqqs 7571	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
6	5	oveq2i 6029	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7		1pi 7535	. . . . 5 ⊢ 1o ∈ N
8		mulpipqqs 7593	. . . . 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 439	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
10	6, 9	eqtrid 2276	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
11		mulcompig 7551	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
12	7, 11	mpan 424	. . . . . 6 ⊢ (𝑥 ∈ N → (1o ·N 𝑥) = (𝑥 ·N 1o))
13	12	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
14		mulcompig 7551	. . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
15	7, 14	mpan 424	. . . . . 6 ⊢ (𝑦 ∈ N → (1o ·N 𝑦) = (𝑦 ·N 1o))
16	15	adantl 277	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
17	13, 16	opeq12d 3870	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
18	17	eceq1d 6738	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19		mulcanenqec 7606	. . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
20	7, 19	mp3an1 1360	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
21	10, 18, 20	3eqtr2d 2270	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q )
22	1, 4, 21	ecoptocl 6791	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ⟨cop 3672  (class class class)co 6018  1_oc1o 6575  [cec 6700  Ncnpi 7492   ·_N cmi 7494   ~_Q ceq 7499  Qcnq 7500  1_Qc1q 7501   ·_Q cmq 7503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-mi 7526  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-mqqs 7570  df-1nqqs 7571

This theorem is referenced by:  recmulnqg  7611  rec1nq  7615  ltaddnq  7627  halfnqq  7630  prarloclemarch  7638  ltrnqg  7640  addnqprllem  7747  addnqprulem  7748  addnqprl  7749  addnqpru  7750  appdivnq  7783  prmuloc2  7787  mulnqprl  7788  mulnqpru  7789  1idprl  7810  1idpru  7811  recexprlem1ssl  7853  recexprlem1ssu  7854