Theorem mulidnq 10954

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 10919	. . 3 ⊢ 1Q ∈ Q
2		mulpqnq 10932	. . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
3	1, 2	mpan2 689	. 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4		relxp 5693	. . . . . . 7 ⊢ Rel (N × N)
5		elpqn 10916	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
6		1st2nd 8021	. . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
7	4, 5, 6	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8		df-1nq 10907	. . . . . . 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 8003	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
12	5, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
13		xp2nd 8004	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
14	5, 13	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
15		1pi 10874	. . . . . . 7 ⊢ 1o ∈ N
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N)
17		mulpipq 10931	. . . . . 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 837	. . . . 5 ⊢ (𝐴 ∈ Q → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩)
19		mulidpi 10877	. . . . . . . 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 10877	. . . . . . . 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 4880	. . . . . 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 2776	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25, 7	eqtr4d 2775	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴)
27	26	fveq2d 6892	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28		nqerid 10924	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
29	3, 27, 28	3eqtrd 2776	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ⟨cop 4633   × cxp 5673  Rel wrel 5680  ‘cfv 6540  (class class class)co 7405  1^st c1st 7969  2^nd c2nd 7970  1_oc1o 8455  Ncnpi 10835   ·_N cmi 10837   ·_pQ cmpq 10840  Qcnq 10843  1_Qc1q 10844  [Q]cerq 10845   ·_Q cmq 10847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8699  df-ni 10863  df-mi 10865  df-lti 10866  df-mpq 10900  df-enq 10902  df-nq 10903  df-erq 10904  df-mq 10906  df-1nq 10907

This theorem is referenced by:  recmulnq  10955  ltaddnq  10965  halfnq  10967  ltrnq  10970  addclprlem1  11007  addclprlem2  11008  mulclprlem  11010  1idpr  11020  prlem934  11024  prlem936  11038  reclem3pr  11040