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| Mirrors > Home > MPE Home > Th. List > grpodivinv | Structured version Visualization version GIF version | ||
| Description: Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grpdiv.1 | β’ π = ran πΊ |
| grpdiv.2 | β’ π = (invβπΊ) |
| grpdiv.3 | β’ π· = ( /π βπΊ) |
| Ref | Expression |
|---|---|
| grpodivinv | β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄π·(πβπ΅)) = (π΄πΊπ΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpdiv.1 | . . . . 5 β’ π = ran πΊ | |
| 2 | grpdiv.2 | . . . . 5 β’ π = (invβπΊ) | |
| 3 | 1, 2 | grpoinvcl 30378 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π) β (πβπ΅) β π) |
| 4 | 3 | 3adant2 1128 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (πβπ΅) β π) |
| 5 | grpdiv.3 | . . . 4 β’ π· = ( /π βπΊ) | |
| 6 | 1, 2, 5 | grpodivval 30389 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π β§ (πβπ΅) β π) β (π΄π·(πβπ΅)) = (π΄πΊ(πβ(πβπ΅)))) |
| 7 | 4, 6 | syld3an3 1406 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄π·(πβπ΅)) = (π΄πΊ(πβ(πβπ΅)))) |
| 8 | 1, 2 | grpo2inv 30385 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π) β (πβ(πβπ΅)) = π΅) |
| 9 | 8 | 3adant2 1128 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (πβ(πβπ΅)) = π΅) |
| 10 | 9 | oveq2d 7432 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄πΊ(πβ(πβπ΅))) = (π΄πΊπ΅)) |
| 11 | 7, 10 | eqtrd 2765 | 1 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄π·(πβπ΅)) = (π΄πΊπ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 ran crn 5673 βcfv 6543 (class class class)co 7416 GrpOpcgr 30343 invcgn 30345 /π cgs 30346 |
| 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-pow 5359 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-pw 4600 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-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-grpo 30347 df-gid 30348 df-ginv 30349 df-gdiv 30350 |
| This theorem is referenced by: ablodivdiv4 30408 |
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