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Mirrors > Home > ILE Home > Th. List > enqer | Unicode 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.) |
Ref | Expression |
---|---|
enqer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 7269 | . 2 | |
2 | mulcompig 7253 | . 2 | |
3 | mulclpi 7250 | . 2 | |
4 | mulasspig 7254 | . 2 | |
5 | mulcanpig 7257 | . . 3 | |
6 | 5 | biimpd 143 | . 2 |
7 | 1, 2, 3, 4, 6 | ecopoverg 6583 | 1 |
Colors of variables: wff set class |
Syntax hints: w3a 963 wceq 1335 wcel 2128 cxp 4586 (class class class)co 5826 wer 6479 cnpi 7194 cmi 7196 ceq 7201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-oadd 6369 df-omul 6370 df-er 6482 df-ni 7226 df-mi 7228 df-enq 7269 |
This theorem is referenced by: enqeceq 7281 0nnq 7286 addpipqqs 7292 mulpipqqs 7295 ordpipqqs 7296 mulcanenqec 7308 |
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