Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6462, 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 7594), 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 7594 | . 2 ⊢ ℂ = (R × R) | |
2 | qsid 6462 | . 2 ⊢ ((R × R) / ◡ E ) = (R × R) | |
3 | 1, 2 | eqtr4i 2141 | 1 ⊢ ℂ = ((R × R) / ◡ E ) |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 E cep 4179 × cxp 4507 ◡ccnv 4508 / cqs 6396 Rcnr 7073 ℂcc 7586 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-eprel 4181 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-ec 6399 df-qs 6403 df-c 7594 |
This theorem is referenced by: axmulcom 7647 axaddass 7648 axmulass 7649 axdistr 7650 |
Copyright terms: Public domain | W3C validator |