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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 29508 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβπ΄) β π) |
4 | eqid 2733 | . . . . 5 β’ (GIdβπΊ) = (GIdβπΊ) | |
5 | 1, 4, 2 | grporinv 29511 | . . . 4 β’ ((πΊ β GrpOp β§ (πβπ΄) β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = (GIdβπΊ)) |
6 | 3, 5 | syldan 592 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = (GIdβπΊ)) |
7 | 1, 4, 2 | grpolinv 29510 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊπ΄) = (GIdβπΊ)) |
8 | 6, 7 | eqtr4d 2776 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β ((πβπ΄)πΊ(πβ(πβπ΄))) = ((πβπ΄)πΊπ΄)) |
9 | 1, 2 | grpoinvcl 29508 | . . . . 5 β’ ((πΊ β GrpOp β§ (πβπ΄) β π) β (πβ(πβπ΄)) β π) |
10 | 3, 9 | syldan 592 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β (πβ(πβπ΄)) β π) |
11 | simpr 486 | . . . 4 β’ ((πΊ β GrpOp β§ π΄ β π) β π΄ β π) | |
12 | 10, 11, 3 | 3jca 1129 | . . 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 29514 | . . 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 592 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 ran crn 5635 βcfv 6497 (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: grpoinvf 29516 grpodivinv 29520 grpoinvdiv 29521 nvnegneg 29633 |
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