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Mirrors > Home > MPE Home > Th. List > enrer | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
enrer | ⊢ ~R Er (P × P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enr 9877 | . 2 ⊢ ~R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} | |
2 | addcompr 9843 | . 2 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥) | |
3 | addclpr 9840 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P 𝑦) ∈ P) | |
4 | addasspr 9844 | . 2 ⊢ ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧)) | |
5 | addcanpr 9868 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 𝑦) = (𝑥 +P 𝑧) → 𝑦 = 𝑧)) | |
6 | 1, 2, 3, 4, 5 | ecopover 7851 | 1 ⊢ ~R Er (P × P) |
Colors of variables: wff setvar class |
Syntax hints: × cxp 5112 Er wer 7739 Pcnp 9681 +P cpp 9683 ~R cer 9686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-plp 9805 df-ltp 9807 df-enr 9877 |
This theorem is referenced by: enreceq 9887 prsrlem1 9893 addsrmo 9894 mulsrmo 9895 ltsrpr 9898 0nsr 9900 axcnex 9968 wuncn 9991 |
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