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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 30388 | . . . 4 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
5 | 4 | oveqd 7433 | . . 3 β’ (πΊ β GrpOp β (π΄π·π΅) = (π΄(π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))π΅)) |
6 | oveq1 7423 | . . . 4 β’ (π₯ = π΄ β (π₯πΊ(πβπ¦)) = (π΄πΊ(πβπ¦))) | |
7 | fveq2 6892 | . . . . 5 β’ (π¦ = π΅ β (πβπ¦) = (πβπ΅)) | |
8 | 7 | oveq2d 7432 | . . . 4 β’ (π¦ = π΅ β (π΄πΊ(πβπ¦)) = (π΄πΊ(πβπ΅))) |
9 | eqid 2725 | . . . 4 β’ (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) | |
10 | ovex 7449 | . . . 4 β’ (π΄πΊ(πβπ΅)) β V | |
11 | 6, 8, 9, 10 | ovmpo 7578 | . . 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 2785 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 ran crn 5673 βcfv 6543 (class class class)co 7416 β cmpo 7418 GrpOpcgr 30343 invcgn 30345 /π cgs 30346 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-gdiv 30350 |
This theorem is referenced by: grpodivinv 30390 grpoinvdiv 30391 grpodivdiv 30394 grpomuldivass 30395 grpodivid 30396 grponpcan 30397 ablodivdiv4 30408 nvmval 30496 rngosub 37460 |
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