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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 6558, 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 7751), 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 7751 | . 2 | |
2 | qsid 6558 | . 2 | |
3 | 1, 2 | eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1342 cep 4260 cxp 4597 ccnv 4598 cqs 6492 cnr 7230 cc 7743 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-sbc 2948 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-eprel 4262 df-xp 4605 df-cnv 4607 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-ec 6495 df-qs 6499 df-c 7751 |
This theorem is referenced by: axmulcom 7804 axaddass 7805 axmulass 7806 axdistr 7807 |
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