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Theorem enrer 9645
Description: The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
enrer ~R Er (P × P)

Proof of Theorem enrer
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enr 9636 . 2 ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
2 addcompr 9602 . 2 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
3 addclpr 9599 . 2 ((𝑥P𝑦P) → (𝑥 +P 𝑦) ∈ P)
4 addasspr 9603 . 2 ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
5 addcanpr 9627 . 2 ((𝑥P𝑦P) → ((𝑥 +P 𝑦) = (𝑥 +P 𝑧) → 𝑦 = 𝑧))
61, 2, 3, 4, 5ecopover 7618 1 ~R Er (P × P)
Colors of variables: wff setvar class
Syntax hints:   × cxp 4930   Er wer 7506  Pcnp 9440   +P cpp 9442   ~R cer 9445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728  ax-inf2 8301
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-ov 6434  df-oprab 6435  df-mpt2 6436  df-om 6839  df-1st 6939  df-2nd 6940  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-1o 7327  df-oadd 7331  df-omul 7332  df-er 7509  df-ni 9453  df-pli 9454  df-mi 9455  df-lti 9456  df-plpq 9489  df-mpq 9490  df-ltpq 9491  df-enq 9492  df-nq 9493  df-erq 9494  df-plq 9495  df-mq 9496  df-1nq 9497  df-rq 9498  df-ltnq 9499  df-np 9562  df-plp 9564  df-ltp 9566  df-enr 9636
This theorem is referenced by:  enreceq  9646  prsrlem1  9652  addsrmo  9653  mulsrmo  9654  ltsrpr  9657  0nsr  9659  axcnex  9727  wuncn  9750
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