![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpoinvval | Structured version Visualization version GIF version |
Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinvfval.1 | β’ π = ran πΊ |
grpinvfval.2 | β’ π = (GIdβπΊ) |
grpinvfval.3 | β’ π = (invβπΊ) |
Ref | Expression |
---|---|
grpoinvval | β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) = (β©π¦ β π (π¦πΊπ΄) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvfval.1 | . . . 4 β’ π = ran πΊ | |
2 | grpinvfval.2 | . . . 4 β’ π = (GIdβπΊ) | |
3 | grpinvfval.3 | . . . 4 β’ π = (invβπΊ) | |
4 | 1, 2, 3 | grpoinvfval 30376 | . . 3 β’ (πΊ β GrpOp β π = (π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π))) |
5 | 4 | fveq1d 6894 | . 2 β’ (πΊ β GrpOp β (πβπ΄) = ((π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π))βπ΄)) |
6 | oveq2 7424 | . . . . 5 β’ (π₯ = π΄ β (π¦πΊπ₯) = (π¦πΊπ΄)) | |
7 | 6 | eqeq1d 2727 | . . . 4 β’ (π₯ = π΄ β ((π¦πΊπ₯) = π β (π¦πΊπ΄) = π)) |
8 | 7 | riotabidv 7374 | . . 3 β’ (π₯ = π΄ β (β©π¦ β π (π¦πΊπ₯) = π) = (β©π¦ β π (π¦πΊπ΄) = π)) |
9 | eqid 2725 | . . 3 β’ (π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π)) = (π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π)) | |
10 | riotaex 7376 | . . 3 β’ (β©π¦ β π (π¦πΊπ΄) = π) β V | |
11 | 8, 9, 10 | fvmpt 7000 | . 2 β’ (π΄ β π β ((π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π))βπ΄) = (β©π¦ β π (π¦πΊπ΄) = π)) |
12 | 5, 11 | sylan9eq 2785 | 1 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) = (β©π¦ β π (π¦πΊπ΄) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¦ cmpt 5226 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-ginv 30349 |
This theorem is referenced by: grpoinvcl 30378 grpoinv 30379 |
Copyright terms: Public domain | W3C validator |