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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 | โข ~Q Er (N ร N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 10903 | . 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 10888 | . 2 โข (๐ฅ ยทN ๐ฆ) = (๐ฆ ยทN ๐ฅ) | |
3 | mulclpi 10885 | . 2 โข ((๐ฅ โ N โง ๐ฆ โ N) โ (๐ฅ ยทN ๐ฆ) โ N) | |
4 | mulasspi 10889 | . 2 โข ((๐ฅ ยทN ๐ฆ) ยทN ๐ง) = (๐ฅ ยทN (๐ฆ ยทN ๐ง)) | |
5 | mulcanpi 10892 | . . 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 8812 | 1 โข ~Q Er (N ร N) |
Colors of variables: wff setvar class |
Syntax hints: โง wa 395 = wceq 1533 โ wcel 2098 ร cxp 5665 (class class class)co 7402 Er wer 8697 Ncnpi 10836 ยทN cmi 10838 ~Q ceq 10843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-oadd 8466 df-omul 8467 df-er 8700 df-ni 10864 df-mi 10866 df-enq 10903 |
This theorem is referenced by: nqereu 10921 nqerf 10922 nqerid 10925 enqeq 10926 nqereq 10927 adderpq 10948 mulerpq 10949 1nqenq 10954 |
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