Theorem mulidnq 10875

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 10840	. . 3 ⊢ 1Q ∈ Q
2		mulpqnq 10853	. . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
3	1, 2	mpan2 692	. 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4		relxp 5640	. . . . . . 7 ⊢ Rel (N × N)
5		elpqn 10837	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
6		1st2nd 7983	. . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
7	4, 5, 6	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8		df-1nq 10828	. . . . . . 7 ⊢ 1Q = ⟨1o, 1o⟩
9	8	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = ⟨1o, 1o⟩)
10	7, 9	oveq12d 7376	. . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11		xp1st 7965	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
12	5, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
13		xp2nd 7966	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
14	5, 13	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
15		1pi 10795	. . . . . . 7 ⊢ 1o ∈ N
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N)
17		mulpipq 10852	. . . . . 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 839	. . . . 5 ⊢ (𝐴 ∈ Q → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)⟩)
19		mulidpi 10798	. . . . . . . 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 10798	. . . . . . . 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 4825	. . . . . 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 6836	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28		nqerid 10845	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
29	3, 27, 28	3eqtrd 2776	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   × cxp 5620  Rel wrel 5627  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  1_oc1o 8389  Ncnpi 10756   ·_N cmi 10758   ·_pQ cmpq 10761  Qcnq 10764  1_Qc1q 10765  [Q]cerq 10766   ·_Q cmq 10768

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-oadd 8400  df-omul 8401  df-er 8634  df-ni 10784  df-mi 10786  df-lti 10787  df-mpq 10821  df-enq 10823  df-nq 10824  df-erq 10825  df-mq 10827  df-1nq 10828

This theorem is referenced by:  recmulnq  10876  ltaddnq  10886  halfnq  10888  ltrnq  10891  addclprlem1  10928  addclprlem2  10929  mulclprlem  10931  1idpr  10941  prlem934  10945  prlem936  10959  reclem3pr  10961