Theorem mulidnq 9785

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 9750	. . 3 ⊢ 1Q ∈ Q
2		mulpqnq 9763	. . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
3	1, 2	mpan2 707	. 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4		relxp 5227	. . . . . . 7 ⊢ Rel (N × N)
5		elpqn 9747	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
6		1st2nd 7214	. . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
7	4, 5, 6	sylancr 695	. . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8		df-1nq 9738	. . . . . . 7 ⊢ 1Q = ⟨1𝑜, 1𝑜⟩
9	8	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = ⟨1𝑜, 1𝑜⟩)
10	7, 9	oveq12d 6668	. . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩))
11		xp1st 7198	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
12	5, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
13		xp2nd 7199	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
14	5, 13	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
15		1pi 9705	. . . . . . 7 ⊢ 1𝑜 ∈ N
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ Q → 1𝑜 ∈ N)
17		mulpipq 9762	. . . . . 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 1327	. . . . 5 ⊢ (𝐴 ∈ Q → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st ‘𝐴) ·N 1𝑜), ((2nd ‘𝐴) ·N 1𝑜)⟩)
19		mulidpi 9708	. . . . . . . 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 9708	. . . . . . . 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 4410	. . . . . 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 2660	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25, 7	eqtr4d 2659	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴)
27	26	fveq2d 6195	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28		nqerid 9755	. 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
29	3, 27, 28	3eqtrd 2660	1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   × cxp 5112  Rel wrel 5119  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  1_𝑜c1o 7553  Ncnpi 9666   ·_N cmi 9668   ·_pQ cmpq 9671  Qcnq 9674  1_Qc1q 9675  [Q]cerq 9676   ·_Q cmq 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-mpq 9731  df-enq 9733  df-nq 9734  df-erq 9735  df-mq 9737  df-1nq 9738

This theorem is referenced by:  recmulnq  9786  ltaddnq  9796  halfnq  9798  ltrnq  9801  addclprlem1  9838  addclprlem2  9839  mulclprlem  9841  1idpr  9851  prlem934  9855  prlem936  9869  reclem3pr  9871