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Mirrors > Home > MPE Home > Th. List > grpo2inv | Structured version Visualization version GIF version |
Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpasscan1.1 | β’ π = ran πΊ |
grpasscan1.2 | β’ π = (invβπΊ) |
Ref | Expression |
---|---|
grpo2inv | β’ ((πΊ β GrpOp β§ π΄ β π) β (πβ(πβπ΄)) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpasscan1.1 | . . . . 5 β’ π = ran πΊ | |
2 | grpasscan1.2 | . . . . 5 β’ π = (invβπΊ) | |
3 | 1, 2 | grpoinvcl 30286 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) β π) |
4 | eqid 2726 | . . . . 5 β’ (GIdβπΊ) = (GIdβπΊ) | |
5 | 1, 4, 2 | grporinv 30289 | . . . 4 β’ ((πΊ β GrpOp β§ (πβπ΄) β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = (GIdβπΊ)) |
6 | 3, 5 | syldan 590 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = (GIdβπΊ)) |
7 | 1, 4, 2 | grpolinv 30288 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊπ΄) = (GIdβπΊ)) |
8 | 6, 7 | eqtr4d 2769 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = ((πβπ΄)πΊπ΄)) |
9 | 1, 2 | grpoinvcl 30286 | . . . . 5 β’ ((πΊ β GrpOp β§ (πβπ΄) β π) β (πβ(πβπ΄)) β π) |
10 | 3, 9 | syldan 590 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβ(πβπ΄)) β π) |
11 | simpr 484 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β π΄ β π) | |
12 | 10, 11, 3 | 3jca 1125 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβ(πβπ΄)) β π β§ π΄ β π β§ (πβπ΄) β π)) |
13 | 1 | grpolcan 30292 | . . 3 β’ ((πΊ β GrpOp β§ ((πβ(πβπ΄)) β π β§ π΄ β π β§ (πβπ΄) β π)) β (((πβπ΄)πΊ(πβ(πβπ΄))) = ((πβπ΄)πΊπ΄) β (πβ(πβπ΄)) = π΄)) |
14 | 12, 13 | syldan 590 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (((πβπ΄)πΊ(πβ(πβπ΄))) = ((πβπ΄)πΊπ΄) β (πβ(πβπ΄)) = π΄)) |
15 | 8, 14 | mpbid 231 | 1 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβ(πβπ΄)) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 ran crn 5670 βcfv 6537 (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-grpo 30255 df-gid 30256 df-ginv 30257 |
This theorem is referenced by: grpoinvf 30294 grpodivinv 30298 grpoinvdiv 30299 nvnegneg 30411 |
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