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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 2728 | . . . . . . . 8 β’ (invβπΊ) = (invβπΊ) | |
| 3 | 1, 2 | grpoinvcl 30362 | . . . . . . 7 β’ ((πΊ β GrpOp β§ π¦ β π) β ((invβπΊ)βπ¦) β π) |
| 4 | 3 | 3adant2 1128 | . . . . . 6 β’ ((πΊ β GrpOp β§ π₯ β π β§ π¦ β π) β ((invβπΊ)βπ¦) β π) |
| 5 | 1 | grpocl 30338 | . . . . . 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 1406 | . . . . 5 β’ ((πΊ β GrpOp β§ π₯ β π β§ π¦ β π) β (π₯πΊ((invβπΊ)βπ¦)) β π) |
| 7 | 6 | 3expib 1119 | . . . 4 β’ (πΊ β GrpOp β ((π₯ β π β§ π¦ β π) β (π₯πΊ((invβπΊ)βπ¦)) β π)) |
| 8 | 7 | ralrimivv 3196 | . . 3 β’ (πΊ β GrpOp β βπ₯ β π βπ¦ β π (π₯πΊ((invβπΊ)βπ¦)) β π) |
| 9 | eqid 2728 | . . . 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 8080 | . . 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 30372 | . . 3 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ((invβπΊ)βπ¦)))) |
| 14 | 13 | feq1d 6712 | . 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 256 | 1 β’ (πΊ β GrpOp β π·:(π Γ π)βΆπ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3058 Γ cxp 5680 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 β cmpo 7428 GrpOpcgr 30327 invcgn 30329 /π cgs 30330 |
| 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-pow 5369 ax-pr 5433 ax-un 7748 |
| 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-pw 4608 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-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-grpo 30331 df-gid 30332 df-ginv 30333 df-gdiv 30334 |
| This theorem is referenced by: grpodivcl 30377 |
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