![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpodivfval | Structured version Visualization version GIF version |
Description: Group division (or subtraction) operation. (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 |
---|---|
grpodivfval | β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdiv.3 | . 2 β’ π· = ( /π βπΊ) | |
2 | grpdiv.1 | . . . . 5 β’ π = ran πΊ | |
3 | rnexg 7833 | . . . . 5 β’ (πΊ β GrpOp β ran πΊ β V) | |
4 | 2, 3 | eqeltrid 2842 | . . . 4 β’ (πΊ β GrpOp β π β V) |
5 | mpoexga 8002 | . . . 4 β’ ((π β V β§ π β V) β (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) β V) | |
6 | 4, 4, 5 | syl2anc 584 | . . 3 β’ (πΊ β GrpOp β (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) β V) |
7 | rneq 5889 | . . . . . 6 β’ (π = πΊ β ran π = ran πΊ) | |
8 | 7, 2 | eqtr4di 2795 | . . . . 5 β’ (π = πΊ β ran π = π) |
9 | id 22 | . . . . . 6 β’ (π = πΊ β π = πΊ) | |
10 | eqidd 2738 | . . . . . 6 β’ (π = πΊ β π₯ = π₯) | |
11 | fveq2 6839 | . . . . . . . 8 β’ (π = πΊ β (invβπ) = (invβπΊ)) | |
12 | grpdiv.2 | . . . . . . . 8 β’ π = (invβπΊ) | |
13 | 11, 12 | eqtr4di 2795 | . . . . . . 7 β’ (π = πΊ β (invβπ) = π) |
14 | 13 | fveq1d 6841 | . . . . . 6 β’ (π = πΊ β ((invβπ)βπ¦) = (πβπ¦)) |
15 | 9, 10, 14 | oveq123d 7372 | . . . . 5 β’ (π = πΊ β (π₯π((invβπ)βπ¦)) = (π₯πΊ(πβπ¦))) |
16 | 8, 8, 15 | mpoeq123dv 7426 | . . . 4 β’ (π = πΊ β (π₯ β ran π, π¦ β ran π β¦ (π₯π((invβπ)βπ¦))) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
17 | df-gdiv 29267 | . . . 4 β’ /π = (π β GrpOp β¦ (π₯ β ran π, π¦ β ran π β¦ (π₯π((invβπ)βπ¦)))) | |
18 | 16, 17 | fvmptg 6943 | . . 3 β’ ((πΊ β GrpOp β§ (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) β V) β ( /π βπΊ) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
19 | 6, 18 | mpdan 685 | . 2 β’ (πΊ β GrpOp β ( /π βπΊ) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
20 | 1, 19 | eqtrid 2789 | 1 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3443 ran crn 5632 βcfv 6493 (class class class)co 7351 β cmpo 7353 GrpOpcgr 29260 invcgn 29262 /π cgs 29263 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-gdiv 29267 |
This theorem is referenced by: grpodivval 29306 grpodivf 29309 nvmfval 29415 |
Copyright terms: Public domain | W3C validator |