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

Theorem enqer 10964
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 10954 . 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 10939 . 2 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
3 mulclpi 10936 . 2 ((𝑥N𝑦N) → (𝑥 ·N 𝑦) ∈ N)
4 mulasspi 10940 . 2 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5 mulcanpi 10943 . . 3 ((𝑥N𝑦N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) ↔ 𝑦 = 𝑧))
65biimpd 228 . 2 ((𝑥N𝑦N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) → 𝑦 = 𝑧))
71, 2, 3, 4, 6ecopover 8850 1 ~Q Er (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wcel 2099   × cxp 5680  (class class class)co 7424   Er wer 8731  Ncnpi 10887   ·N cmi 10889   ~Q ceq 10894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-oadd 8500  df-omul 8501  df-er 8734  df-ni 10915  df-mi 10917  df-enq 10954
This theorem is referenced by:  nqereu  10972  nqerf  10973  nqerid  10976  enqeq  10977  nqereq  10978  adderpq  10999  mulerpq  11000  1nqenq  11005
  Copyright terms: Public domain W3C validator