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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 10322 | . 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 10307 | . 2 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥) | |
3 | mulclpi 10304 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (𝑥 ·N 𝑦) ∈ N) | |
4 | mulasspi 10308 | . 2 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧)) | |
5 | mulcanpi 10311 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) ↔ 𝑦 = 𝑧)) | |
6 | 5 | biimpd 232 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) → 𝑦 = 𝑧)) |
7 | 1, 2, 3, 4, 6 | ecopover 8384 | 1 ⊢ ~Q Er (N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 × cxp 5517 (class class class)co 7135 Er wer 8269 Ncnpi 10255 ·N cmi 10257 ~Q ceq 10262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-omul 8090 df-er 8272 df-ni 10283 df-mi 10285 df-enq 10322 |
This theorem is referenced by: nqereu 10340 nqerf 10341 nqerid 10344 enqeq 10345 nqereq 10346 adderpq 10367 mulerpq 10368 1nqenq 10373 |
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