Theorem mulidnq 10650

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 10615	. . 3 ⊢ 1Q ∈ Q
2		mulpqnq 10628	. . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
3	1, 2	mpan2 687	. 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4		relxp 5598	. . . . . . 7 ⊢ Rel (N × N)
5		elpqn 10612	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
6		1st2nd 7853	. . . . . . 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 10603	. . . . . . 7 ⊢ 1Q = ⟨1o, 1o⟩
9	8	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = ⟨1o, 1o⟩)
10	7, 9	oveq12d 7273	. . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11		xp1st 7836	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
12	5, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
13		xp2nd 7837	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
14	5, 13	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
15		1pi 10570	. . . . . . 7 ⊢ 1o ∈ N
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N)
17		mulpipq 10627	. . . . . 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 835	. . . . 5 ⊢ (𝐴 ∈ Q → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩)
19		mulidpi 10573	. . . . . . . 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 10573	. . . . . . . 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 4809	. . . . . 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 2782	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25, 7	eqtr4d 2781	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴)
27	26	fveq2d 6760	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28		nqerid 10620	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
29	3, 27, 28	3eqtrd 2782	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   × cxp 5578  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  1_oc1o 8260  Ncnpi 10531   ·_N cmi 10533   ·_pQ cmpq 10536  Qcnq 10539  1_Qc1q 10540  [Q]cerq 10541   ·_Q cmq 10543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ni 10559  df-mi 10561  df-lti 10562  df-mpq 10596  df-enq 10598  df-nq 10599  df-erq 10600  df-mq 10602  df-1nq 10603

This theorem is referenced by:  recmulnq  10651  ltaddnq  10661  halfnq  10663  ltrnq  10666  addclprlem1  10703  addclprlem2  10704  mulclprlem  10706  1idpr  10716  prlem934  10720  prlem936  10734  reclem3pr  10736