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Theorem enqer 9946
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 9936 . 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 9921 . 2 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
3 mulclpi 9918 . 2 ((𝑥N𝑦N) → (𝑥 ·N 𝑦) ∈ N)
4 mulasspi 9922 . 2 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5 mulcanpi 9925 . . 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 8005 1 ~Q Er (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145   × cxp 5248  (class class class)co 6794   Er wer 7894  Ncnpi 9869   ·N cmi 9871   ~Q ceq 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-oadd 7718  df-omul 7719  df-er 7897  df-ni 9897  df-mi 9899  df-enq 9936
This theorem is referenced by:  nqereu  9954  nqerf  9955  nqerid  9958  enqeq  9959  nqereq  9960  adderpq  9981  mulerpq  9982  1nqenq  9987
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