![]() |
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 | โข ~Q Er (N ร N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 10934 | . 2 โข ~Q = {โจ๐ฅ, ๐ฆโฉ โฃ ((๐ฅ โ (N ร N) โง ๐ฆ โ (N ร N)) โง โ๐งโ๐คโ๐ฃโ๐ข((๐ฅ = โจ๐ง, ๐คโฉ โง ๐ฆ = โจ๐ฃ, ๐ขโฉ) โง (๐ง ยทN ๐ข) = (๐ค ยทN ๐ฃ)))} | |
2 | mulcompi 10919 | . 2 โข (๐ฅ ยทN ๐ฆ) = (๐ฆ ยทN ๐ฅ) | |
3 | mulclpi 10916 | . 2 โข ((๐ฅ โ N โง ๐ฆ โ N) โ (๐ฅ ยทN ๐ฆ) โ N) | |
4 | mulasspi 10920 | . 2 โข ((๐ฅ ยทN ๐ฆ) ยทN ๐ง) = (๐ฅ ยทN (๐ฆ ยทN ๐ง)) | |
5 | mulcanpi 10923 | . . 3 โข ((๐ฅ โ N โง ๐ฆ โ N) โ ((๐ฅ ยทN ๐ฆ) = (๐ฅ ยทN ๐ง) โ ๐ฆ = ๐ง)) | |
6 | 5 | biimpd 228 | . 2 โข ((๐ฅ โ N โง ๐ฆ โ N) โ ((๐ฅ ยทN ๐ฆ) = (๐ฅ ยทN ๐ง) โ ๐ฆ = ๐ง)) |
7 | 1, 2, 3, 4, 6 | ecopover 8839 | 1 โข ~Q Er (N ร N) |
Colors of variables: wff setvar class |
Syntax hints: โง wa 395 = wceq 1534 โ wcel 2099 ร cxp 5676 (class class class)co 7420 Er wer 8721 Ncnpi 10867 ยทN cmi 10869 ~Q ceq 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-oadd 8490 df-omul 8491 df-er 8724 df-ni 10895 df-mi 10897 df-enq 10934 |
This theorem is referenced by: nqereu 10952 nqerf 10953 nqerid 10956 enqeq 10957 nqereq 10958 adderpq 10979 mulerpq 10980 1nqenq 10985 |
Copyright terms: Public domain | W3C validator |