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| Mirrors > Home > MPE Home > Th. List > grpodivdiv | Structured version Visualization version GIF version | ||
| Description: Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grpdivf.1 | β’ π = ran πΊ |
| grpdivf.3 | β’ π· = ( /π βπΊ) |
| Ref | Expression |
|---|---|
| grpodivdiv | β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·(π΅π·πΆ)) = (π΄πΊ(πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 483 | . . 3 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β GrpOp) | |
| 2 | simpr1 1194 | . . 3 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 3 | grpdivf.1 | . . . . 5 β’ π = ran πΊ | |
| 4 | grpdivf.3 | . . . . 5 β’ π· = ( /π βπΊ) | |
| 5 | 3, 4 | grpodivcl 29779 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π β§ πΆ β π) β (π΅π·πΆ) β π) |
| 6 | 5 | 3adant3r1 1182 | . . 3 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅π·πΆ) β π) |
| 7 | eqid 2732 | . . . 4 β’ (invβπΊ) = (invβπΊ) | |
| 8 | 3, 7, 4 | grpodivval 29775 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π β§ (π΅π·πΆ) β π) β (π΄π·(π΅π·πΆ)) = (π΄πΊ((invβπΊ)β(π΅π·πΆ)))) |
| 9 | 1, 2, 6, 8 | syl3anc 1371 | . 2 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·(π΅π·πΆ)) = (π΄πΊ((invβπΊ)β(π΅π·πΆ)))) |
| 10 | 3, 7, 4 | grpoinvdiv 29777 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π β§ πΆ β π) β ((invβπΊ)β(π΅π·πΆ)) = (πΆπ·π΅)) |
| 11 | 10 | 3adant3r1 1182 | . . 3 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((invβπΊ)β(π΅π·πΆ)) = (πΆπ·π΅)) |
| 12 | 11 | oveq2d 7421 | . 2 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄πΊ((invβπΊ)β(π΅π·πΆ))) = (π΄πΊ(πΆπ·π΅))) |
| 13 | 9, 12 | eqtrd 2772 | 1 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·(π΅π·πΆ)) = (π΄πΊ(πΆπ·π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 ran crn 5676 βcfv 6540 (class class class)co 7405 GrpOpcgr 29729 invcgn 29731 /π cgs 29732 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 |
| This theorem is referenced by: ablodivdiv 29793 |
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