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Mirrors > Home > ILE Home > Th. List > dfcnqs | Unicode version |
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6194, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that is a quotient set, even though it is not (compare df-c 6987), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
dfcnqs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 6987 | . 2 | |
2 | qsid 6194 | . 2 | |
3 | 1, 2 | eqtr4i 2104 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1284 cep 4042 cxp 4361 ccnv 4362 cqs 6128 cnr 6487 cc 6979 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-eprel 4044 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-ec 6131 df-qs 6135 df-c 6987 |
This theorem is referenced by: axmulcom 7037 axaddass 7038 axmulass 7039 axdistr 7040 |
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