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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 2728 | . . 3 β’ (2nd βπ ) = (2nd βπ ) | |
3 | eqid 2728 | . . 3 β’ ran πΊ = ran πΊ | |
4 | 1, 2, 3 | rngoi 37405 | . 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 766 | 1 β’ (π β RingOps β πΊ β AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 Γ cxp 5680 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 AbelOpcablo 30374 RingOpscrngo 37400 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-1st 7999 df-2nd 8000 df-rngo 37401 |
This theorem is referenced by: rngoablo2 37415 rngogrpo 37416 rngocom 37419 rngoa32 37421 rngoa4 37422 |
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