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Mirrors > Home > MPE Home > Th. List > grpoinvcl | Structured version Visualization version GIF version |
Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinvcl.1 | β’ π = ran πΊ |
grpinvcl.2 | β’ π = (invβπΊ) |
Ref | Expression |
---|---|
grpoinvcl | β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvcl.1 | . . 3 β’ π = ran πΊ | |
2 | eqid 2725 | . . 3 β’ (GIdβπΊ) = (GIdβπΊ) | |
3 | grpinvcl.2 | . . 3 β’ π = (invβπΊ) | |
4 | 1, 2, 3 | grpoinvval 30377 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) = (β©π¦ β π (π¦πΊπ΄) = (GIdβπΊ))) |
5 | 1, 2 | grpoinveu 30373 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β β!π¦ β π (π¦πΊπ΄) = (GIdβπΊ)) |
6 | riotacl 7390 | . . 3 β’ (β!π¦ β π (π¦πΊπ΄) = (GIdβπΊ) β (β©π¦ β π (π¦πΊπ΄) = (GIdβπΊ)) β π) | |
7 | 5, 6 | syl 17 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (β©π¦ β π (π¦πΊπ΄) = (GIdβπΊ)) β π) |
8 | 4, 7 | eqeltrd 2825 | 1 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β!wreu 3362 ran crn 5673 βcfv 6543 β©crio 7371 (class class class)co 7416 GrpOpcgr 30343 GIdcgi 30344 invcgn 30345 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-grpo 30347 df-gid 30348 df-ginv 30349 |
This theorem is referenced by: grpoinvid1 30382 grpoinvid2 30383 grpolcan 30384 grpo2inv 30385 grpoinvf 30386 grpoinvop 30387 grpodivinv 30390 grpoinvdiv 30391 grpodivf 30392 grpomuldivass 30395 grponpcan 30397 ablodivdiv4 30408 vcm 30430 rngonegcl 37457 isdrngo2 37488 |
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