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Mirrors > Home > ILE Home > Th. List > dfcnqs | GIF version |
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 6494, 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 ℂ is a quotient set, even though it is not (compare df-c 7626), 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 | ⊢ ℂ = ((R × R) / ◡ E ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 7626 | . 2 ⊢ ℂ = (R × R) | |
2 | qsid 6494 | . 2 ⊢ ((R × R) / ◡ E ) = (R × R) | |
3 | 1, 2 | eqtr4i 2163 | 1 ⊢ ℂ = ((R × R) / ◡ E ) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 E cep 4209 × cxp 4537 ◡ccnv 4538 / cqs 6428 Rcnr 7105 ℂcc 7618 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-eprel 4211 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 df-qs 6435 df-c 7626 |
This theorem is referenced by: axmulcom 7679 axaddass 7680 axmulass 7681 axdistr 7682 |
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