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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 7813, 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 9942), 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 9942 | . 2 ⊢ ℂ = (R × R) | |
2 | qsid 7813 | . 2 ⊢ ((R × R) / ◡ E ) = (R × R) | |
3 | 1, 2 | eqtr4i 2647 | 1 ⊢ ℂ = ((R × R) / ◡ E ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 E cep 5028 × cxp 5112 ◡ccnv 5113 / cqs 7741 Rcnr 9687 ℂcc 9934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 df-qs 7748 df-c 9942 |
This theorem is referenced by: axmulcom 9976 axaddass 9977 axmulass 9978 axdistr 9979 |
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