![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > enqer | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
enqer | ⊢ ~Q Er (N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 9771 | . 2 ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | |
2 | mulcompi 9756 | . 2 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥) | |
3 | mulclpi 9753 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (𝑥 ·N 𝑦) ∈ N) | |
4 | mulasspi 9757 | . 2 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧)) | |
5 | mulcanpi 9760 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) ↔ 𝑦 = 𝑧)) | |
6 | 5 | biimpd 219 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) → 𝑦 = 𝑧)) |
7 | 1, 2, 3, 4, 6 | ecopover 7894 | 1 ⊢ ~Q Er (N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∈ wcel 2030 × cxp 5141 (class class class)co 6690 Er wer 7784 Ncnpi 9704 ·N cmi 9706 ~Q ceq 9711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-oadd 7609 df-omul 7610 df-er 7787 df-ni 9732 df-mi 9734 df-enq 9771 |
This theorem is referenced by: nqereu 9789 nqerf 9790 nqerid 9793 enqeq 9794 nqereq 9795 adderpq 9816 mulerpq 9817 1nqenq 9822 |
Copyright terms: Public domain | W3C validator |