MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  enqer Structured version   Visualization version   GIF version

Theorem enqer 9688
Description: The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
enqer ~Q Er (N × N)

Proof of Theorem enqer
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enq 9678 . 2 ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
2 mulcompi 9663 . 2 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
3 mulclpi 9660 . 2 ((𝑥N𝑦N) → (𝑥 ·N 𝑦) ∈ N)
4 mulasspi 9664 . 2 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5 mulcanpi 9667 . . 3 ((𝑥N𝑦N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) ↔ 𝑦 = 𝑧))
65biimpd 219 . 2 ((𝑥N𝑦N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) → 𝑦 = 𝑧))
71, 2, 3, 4, 6ecopover 7797 1 ~Q Er (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1992   × cxp 5077  (class class class)co 6605   Er wer 7685  Ncnpi 9611   ·N cmi 9613   ~Q ceq 9618
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510  df-omul 7511  df-er 7688  df-ni 9639  df-mi 9641  df-enq 9678
This theorem is referenced by:  nqereu  9696  nqerf  9697  nqerid  9700  enqeq  9701  nqereq  9702  adderpq  9723  mulerpq  9724  1nqenq  9729
  Copyright terms: Public domain W3C validator