![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpodivval | Structured version Visualization version GIF version |
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (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 |
---|---|
grpodivval | β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (π΄πΊ(πβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdiv.1 | . . . . 5 β’ π = ran πΊ | |
2 | grpdiv.2 | . . . . 5 β’ π = (invβπΊ) | |
3 | grpdiv.3 | . . . . 5 β’ π· = ( /π βπΊ) | |
4 | 1, 2, 3 | grpodivfval 30296 | . . . 4 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
5 | 4 | oveqd 7422 | . . 3 β’ (πΊ β GrpOp β (π΄π·π΅) = (π΄(π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))π΅)) |
6 | oveq1 7412 | . . . 4 β’ (π₯ = π΄ β (π₯πΊ(πβπ¦)) = (π΄πΊ(πβπ¦))) | |
7 | fveq2 6885 | . . . . 5 β’ (π¦ = π΅ β (πβπ¦) = (πβπ΅)) | |
8 | 7 | oveq2d 7421 | . . . 4 β’ (π¦ = π΅ β (π΄πΊ(πβπ¦)) = (π΄πΊ(πβπ΅))) |
9 | eqid 2726 | . . . 4 β’ (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) | |
10 | ovex 7438 | . . . 4 β’ (π΄πΊ(πβπ΅)) β V | |
11 | 6, 8, 9, 10 | ovmpo 7564 | . . 3 β’ ((π΄ β π β§ π΅ β π) β (π΄(π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))π΅) = (π΄πΊ(πβπ΅))) |
12 | 5, 11 | sylan9eq 2786 | . 2 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) = (π΄πΊ(πβπ΅))) |
13 | 12 | 3impb 1112 | 1 β’ ((πΊ β GrpOp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (π΄πΊ(πβπ΅))) |
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 β cmpo 7407 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-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-gdiv 30258 |
This theorem is referenced by: grpodivinv 30298 grpoinvdiv 30299 grpodivdiv 30302 grpomuldivass 30303 grpodivid 30304 grponpcan 30305 ablodivdiv4 30316 nvmval 30404 rngosub 37311 |
Copyright terms: Public domain | W3C validator |