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| Mirrors > Home > MPE Home > Th. List > grpodivf | Structured version Visualization version GIF version | ||
| Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grpdivf.1 | β’ π = ran πΊ |
| grpdivf.3 | β’ π· = ( /π βπΊ) |
| Ref | Expression |
|---|---|
| grpodivf | β’ (πΊ β GrpOp β π·:(π Γ π)βΆπ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpdivf.1 | . . . . . . . 8 β’ π = ran πΊ | |
| 2 | eqid 2733 | . . . . . . . 8 β’ (invβπΊ) = (invβπΊ) | |
| 3 | 1, 2 | grpoinvcl 29777 | . . . . . . 7 β’ ((πΊ β GrpOp β§ π¦ β π) β ((invβπΊ)βπ¦) β π) |
| 4 | 3 | 3adant2 1132 | . . . . . 6 β’ ((πΊ β GrpOp β§ π₯ β π β§ π¦ β π) β ((invβπΊ)βπ¦) β π) |
| 5 | 1 | grpocl 29753 | . . . . . 6 β’ ((πΊ β GrpOp β§ π₯ β π β§ ((invβπΊ)βπ¦) β π) β (π₯πΊ((invβπΊ)βπ¦)) β π) |
| 6 | 4, 5 | syld3an3 1410 | . . . . 5 β’ ((πΊ β GrpOp β§ π₯ β π β§ π¦ β π) β (π₯πΊ((invβπΊ)βπ¦)) β π) |
| 7 | 6 | 3expib 1123 | . . . 4 β’ (πΊ β GrpOp β ((π₯ β π β§ π¦ β π) β (π₯πΊ((invβπΊ)βπ¦)) β π)) |
| 8 | 7 | ralrimivv 3199 | . . 3 β’ (πΊ β GrpOp β βπ₯ β π βπ¦ β π (π₯πΊ((invβπΊ)βπ¦)) β π) |
| 9 | eqid 2733 | . . . 4 β’ (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦))) = (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦))) | |
| 10 | 9 | fmpo 8054 | . . 3 β’ (βπ₯ β π βπ¦ β π (π₯πΊ((invβπΊ)βπ¦)) β π β (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦))):(π Γ π)βΆπ) |
| 11 | 8, 10 | sylib 217 | . 2 β’ (πΊ β GrpOp β (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦))):(π Γ π)βΆπ) |
| 12 | grpdivf.3 | . . . 4 β’ π· = ( /π βπΊ) | |
| 13 | 1, 2, 12 | grpodivfval 29787 | . . 3 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦)))) |
| 14 | 13 | feq1d 6703 | . 2 β’ (πΊ β GrpOp β (π·:(π Γ π)βΆπ β (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦))):(π Γ π)βΆπ)) |
| 15 | 11, 14 | mpbird 257 | 1 β’ (πΊ β GrpOp β π·:(π Γ π)βΆπ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3062 Γ cxp 5675 ran crn 5678 βΆwf 6540 βcfv 6544 (class class class)co 7409 β cmpo 7411 GrpOpcgr 29742 invcgn 29744 /π cgs 29745 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 |
| This theorem is referenced by: grpodivcl 29792 |
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