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Theorem mulidnq 9745
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 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)

Proof of Theorem mulidnq
StepHypRef Expression
1 1nq 9710 . . 3 1QQ
2 mulpqnq 9723 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 706 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5198 . . . . . . 7 Rel (N × N)
5 elpqn 9707 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7174 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 694 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 9698 . . . . . . 7 1Q = ⟨1𝑜, 1𝑜
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1𝑜, 1𝑜⟩)
107, 9oveq12d 6633 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩))
11 xp1st 7158 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7159 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 9665 . . . . . . 7 1𝑜N
1615a1i 11 . . . . . 6 (𝐴Q → 1𝑜N)
17 mulpipq 9722 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1𝑜N ∧ 1𝑜N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
1812, 14, 16, 16, 17syl22anc 1324 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
19 mulidpi 9668 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
21 mulidpi 9668 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2320, 22opeq12d 4385 . . . . . 6 (𝐴 ∈ (N × N) → ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩ = ⟨(1st𝐴), (2nd𝐴)⟩)
245, 23syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩ = ⟨(1st𝐴), (2nd𝐴)⟩)
2510, 18, 243eqtrd 2659 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2658 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6162 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 9715 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2659 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cop 4161   × cxp 5082  Rel wrel 5089  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  1𝑜c1o 7513  Ncnpi 9626   ·N cmi 9628   ·pQ cmpq 9631  Qcnq 9634  1Qc1q 9635  [Q]cerq 9636   ·Q cmq 9638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-mi 9656  df-lti 9657  df-mpq 9691  df-enq 9693  df-nq 9694  df-erq 9695  df-mq 9697  df-1nq 9698
This theorem is referenced by:  recmulnq  9746  ltaddnq  9756  halfnq  9758  ltrnq  9761  addclprlem1  9798  addclprlem2  9799  mulclprlem  9801  1idpr  9811  prlem934  9815  prlem936  9829  reclem3pr  9831
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