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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 482 | . . 3 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β GrpOp) | |
| 2 | simpr1 1191 | . . 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 30301 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π β§ πΆ β π) β (π΅π·πΆ) β π) |
| 6 | 5 | 3adant3r1 1179 | . . 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 2726 | . . . 4 β’ (invβπΊ) = (invβπΊ) | |
| 8 | 3, 7, 4 | grpodivval 30297 | . . 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 1368 | . 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 30299 | . . . 4 β’ ((πΊ β GrpOp β§ π΅ β π β§ πΆ β π) β ((invβπΊ)β(π΅π·πΆ)) = (πΆπ·π΅)) |
| 11 | 10 | 3adant3r1 1179 | . . 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 2766 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 ran crn 5670 βcfv 6537 (class class class)co 7405 GrpOpcgr 30251 invcgn 30253 /π cgs 30254 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-grpo 30255 df-gid 30256 df-ginv 30257 df-gdiv 30258 |
| This theorem is referenced by: ablodivdiv 30315 |
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