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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 30284 | . . 3 β’ (πΊ β GrpOp β π = (π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π))) |
5 | 4 | fveq1d 6887 | . 2 β’ (πΊ β GrpOp β (πβπ΄) = ((π₯ β π β¦ (β©π¦ β π (π¦πΊπ₯) = π))βπ΄)) |
6 | oveq2 7413 | . . . . 5 β’ (π₯ = π΄ β (π¦πΊπ₯) = (π¦πΊπ΄)) | |
7 | 6 | eqeq1d 2728 | . . . 4 β’ (π₯ = π΄ β ((π¦πΊπ₯) = π β (π¦πΊπ΄) = π)) |
8 | 7 | riotabidv 7363 | . . 3 β’ (π₯ = π΄ β (β©π¦ β π (π¦πΊπ₯) = π) = (β©π¦ β π (π¦πΊπ΄) = π)) |
9 | eqid 2726 | . . 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 7365 | . . 3 β’ (β©π¦ β π (π¦πΊπ΄) = π) β V | |
11 | 8, 9, 10 | fvmpt 6992 | . 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 2786 | 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 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 ran crn 5670 βcfv 6537 β©crio 7360 (class class class)co 7405 GrpOpcgr 30251 GIdcgi 30252 invcgn 30253 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
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-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 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-iun 4992 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-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-ginv 30257 |
This theorem is referenced by: grpoinvcl 30286 grpoinv 30287 |
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