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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoablo | Structured version Visualization version GIF version |
Description: A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringabl.1 | β’ πΊ = (1st βπ ) |
Ref | Expression |
---|---|
rngoablo | β’ (π β RingOps β πΊ β AbelOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringabl.1 | . . 3 β’ πΊ = (1st βπ ) | |
2 | eqid 2726 | . . 3 β’ (2nd βπ ) = (2nd βπ ) | |
3 | eqid 2726 | . . 3 β’ ran πΊ = ran πΊ | |
4 | 1, 2, 3 | rngoi 37279 | . 2 β’ (π β RingOps β ((πΊ β AbelOp β§ (2nd βπ ):(ran πΊ Γ ran πΊ)βΆran πΊ) β§ (βπ₯ β ran πΊβπ¦ β ran πΊβπ§ β ran πΊ(((π₯(2nd βπ )π¦)(2nd βπ )π§) = (π₯(2nd βπ )(π¦(2nd βπ )π§)) β§ (π₯(2nd βπ )(π¦πΊπ§)) = ((π₯(2nd βπ )π¦)πΊ(π₯(2nd βπ )π§)) β§ ((π₯πΊπ¦)(2nd βπ )π§) = ((π₯(2nd βπ )π§)πΊ(π¦(2nd βπ )π§))) β§ βπ₯ β ran πΊβπ¦ β ran πΊ((π₯(2nd βπ )π¦) = π¦ β§ (π¦(2nd βπ )π₯) = π¦)))) |
5 | 4 | simplld 765 | 1 β’ (π β RingOps β πΊ β AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 Γ cxp 5667 ran crn 5670 βΆwf 6532 βcfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 AbelOpcablo 30301 RingOpscrngo 37274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-1st 7971 df-2nd 7972 df-rngo 37275 |
This theorem is referenced by: rngoablo2 37289 rngogrpo 37290 rngocom 37293 rngoa32 37295 rngoa4 37296 |
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