Theorem mulidnq 10960

Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

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

Proof of Theorem mulidnq

Step	Hyp	Ref	Expression
1		1nq 10925	. . 3 ⊢ 1Q ∈ Q
2		mulpqnq 10938	. . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
3	1, 2	mpan2 688	. 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4		relxp 5687	. . . . . . 7 ⊢ Rel (N × N)
5		elpqn 10922	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
6		1st2nd 8024	. . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
7	4, 5, 6	sylancr 586	. . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8		df-1nq 10913	. . . . . . 7 ⊢ 1Q = ⟨1o, 1o⟩
9	8	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = ⟨1o, 1o⟩)
10	7, 9	oveq12d 7423	. . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11		xp1st 8006	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
12	5, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
13		xp2nd 8007	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
14	5, 13	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
15		1pi 10880	. . . . . . 7 ⊢ 1o ∈ N
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N)
17		mulpipq 10937	. . . . . 6 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩)
18	12, 14, 16, 16, 17	syl22anc 836	. . . . 5 ⊢ (𝐴 ∈ Q → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩)
19		mulidpi 10883	. . . . . . . 8 ⊢ ((1st ‘𝐴) ∈ N → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
20	11, 19	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
21		mulidpi 10883	. . . . . . . 8 ⊢ ((2nd ‘𝐴) ∈ N → ((2nd ‘𝐴) ·N 1o) = (2nd ‘𝐴))
22	13, 21	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((2nd ‘𝐴) ·N 1o) = (2nd ‘𝐴))
23	20, 22	opeq12d 4876	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩ = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
24	5, 23	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩ = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	10, 18, 24	3eqtrd 2770	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25, 7	eqtr4d 2769	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴)
27	26	fveq2d 6889	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28		nqerid 10930	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
29	3, 27, 28	3eqtrd 2770	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ⟨cop 4629   × cxp 5667  Rel wrel 5674  ‘cfv 6537  (class class class)co 7405  1^st c1st 7972  2^nd c2nd 7973  1_oc1o 8460  Ncnpi 10841   ·_N cmi 10843   ·_pQ cmpq 10846  Qcnq 10849  1_Qc1q 10850  [Q]cerq 10851   ·_Q cmq 10853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-omul 8472  df-er 8705  df-ni 10869  df-mi 10871  df-lti 10872  df-mpq 10906  df-enq 10908  df-nq 10909  df-erq 10910  df-mq 10912  df-1nq 10913

This theorem is referenced by:  recmulnq  10961  ltaddnq  10971  halfnq  10973  ltrnq  10976  addclprlem1  11013  addclprlem2  11014  mulclprlem  11016  1idpr  11026  prlem934  11030  prlem936  11044  reclem3pr  11046