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Theorem enrer 9574
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 9565 . 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 9531 . 2 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
3 addclpr 9528 . 2 ((𝑥P𝑦P) → (𝑥 +P 𝑦) ∈ P)
4 addasspr 9532 . 2 ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
5 addcanpr 9556 . 2 ((𝑥P𝑦P) → ((𝑥 +P 𝑦) = (𝑥 +P 𝑧) → 𝑦 = 𝑧))
61, 2, 3, 4, 5ecopover 7549 1 ~R Er (P × P)
Colors of variables: wff setvar class
Syntax hints:   × cxp 4878   Er wer 7437  Pcnp 9369   +P cpp 9371   ~R cer 9374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-inf2 8231
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-int 4265  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ni 9382  df-pli 9383  df-mi 9384  df-lti 9385  df-plpq 9418  df-mpq 9419  df-ltpq 9420  df-enq 9421  df-nq 9422  df-erq 9423  df-plq 9424  df-mq 9425  df-1nq 9426  df-rq 9427  df-ltnq 9428  df-np 9491  df-plp 9493  df-ltp 9495  df-enr 9565
This theorem is referenced by:  enreceq  9575  prsrlem1  9581  addsrmo  9582  mulsrmo  9583  ltsrpr  9586  0nsr  9588  axcnex  9656  wuncn  9679
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