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Mirrors > Home > MPE Home > Th. List > grporinv | Structured version Visualization version GIF version |
Description: The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinv.1 | β’ π = ran πΊ |
grpinv.2 | β’ π = (GIdβπΊ) |
grpinv.3 | β’ π = (invβπΊ) |
Ref | Expression |
---|---|
grporinv | β’ ((πΊ β GrpOp β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinv.1 | . . 3 β’ π = ran πΊ | |
2 | grpinv.2 | . . 3 β’ π = (GIdβπΊ) | |
3 | grpinv.3 | . . 3 β’ π = (invβπΊ) | |
4 | 1, 2, 3 | grpoinv 30355 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (((πβπ΄)πΊπ΄) = π β§ (π΄πΊ(πβπ΄)) = π)) |
5 | 4 | simprd 494 | 1 β’ ((πΊ β GrpOp β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 ran crn 5683 βcfv 6553 (class class class)co 7426 GrpOpcgr 30319 GIdcgi 30320 invcgn 30321 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-grpo 30323 df-gid 30324 df-ginv 30325 |
This theorem is referenced by: grpoinvid1 30358 grpoinvid2 30359 grpo2inv 30361 grpoinvop 30363 grpodivid 30372 vcm 30406 nvrinv 30481 rngoaddneg1 37434 |
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