Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfcnqs | Structured version Visualization version 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 8352, which shows that the coset of the converse membership 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 10531), 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.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfcnqs | ⊢ ℂ = ((R × R) / ◡ E ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 10531 | . 2 ⊢ ℂ = (R × R) | |
2 | qsid 8352 | . 2 ⊢ ((R × R) / ◡ E ) = (R × R) | |
3 | 1, 2 | eqtr4i 2844 | 1 ⊢ ℂ = ((R × R) / ◡ E ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 E cep 5457 × cxp 5546 ◡ccnv 5547 / cqs 8277 Rcnr 10275 ℂcc 10523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 df-qs 8284 df-c 10531 |
This theorem is referenced by: axmulcom 10565 axaddass 10566 axmulass 10567 axdistr 10568 |
Copyright terms: Public domain | W3C validator |