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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 2733 | . . 3 β’ (GIdβπΊ) = (GIdβπΊ) | |
3 | grpinvcl.2 | . . 3 β’ π = (invβπΊ) | |
4 | 1, 2, 3 | grpoinvval 29507 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) = (β©π¦ β π (π¦πΊπ΄) = (GIdβπΊ))) |
5 | 1, 2 | grpoinveu 29503 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β β!π¦ β π (π¦πΊπ΄) = (GIdβπΊ)) |
6 | riotacl 7332 | . . 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 2834 | 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 397 = wceq 1542 β wcel 2107 β!wreu 3350 ran crn 5635 βcfv 6497 β©crio 7313 (class class class)co 7358 GrpOpcgr 29473 GIdcgi 29474 invcgn 29475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-grpo 29477 df-gid 29478 df-ginv 29479 |
This theorem is referenced by: grpoinvid1 29512 grpoinvid2 29513 grpolcan 29514 grpo2inv 29515 grpoinvf 29516 grpoinvop 29517 grpodivinv 29520 grpoinvdiv 29521 grpodivf 29522 grpomuldivass 29525 grponpcan 29527 ablodivdiv4 29538 vcm 29560 rngonegcl 36432 isdrngo2 36463 |
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