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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 29518 | . . . 4 β’ (πΊ β GrpOp β π· = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))) |
5 | 4 | oveqd 7375 | . . 3 β’ (πΊ β GrpOp β (π΄π·π΅) = (π΄(π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦)))π΅)) |
6 | oveq1 7365 | . . . 4 β’ (π₯ = π΄ β (π₯πΊ(πβπ¦)) = (π΄πΊ(πβπ¦))) | |
7 | fveq2 6843 | . . . . 5 β’ (π¦ = π΅ β (πβπ¦) = (πβπ΅)) | |
8 | 7 | oveq2d 7374 | . . . 4 β’ (π¦ = π΅ β (π΄πΊ(πβπ¦)) = (π΄πΊ(πβπ΅))) |
9 | eqid 2733 | . . . 4 β’ (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) = (π₯ β π, π¦ β π β¦ (π₯πΊ(πβπ¦))) | |
10 | ovex 7391 | . . . 4 β’ (π΄πΊ(πβπ΅)) β V | |
11 | 6, 8, 9, 10 | ovmpo 7516 | . . 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 2793 | . 2 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) = (π΄πΊ(πβπ΅))) |
13 | 12 | 3impb 1116 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 ran crn 5635 βcfv 6497 (class class class)co 7358 β cmpo 7360 GrpOpcgr 29473 invcgn 29475 /π cgs 29476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-gdiv 29480 |
This theorem is referenced by: grpodivinv 29520 grpoinvdiv 29521 grpodivdiv 29524 grpomuldivass 29525 grpodivid 29526 grponpcan 29527 ablodivdiv4 29538 nvmval 29626 rngosub 36435 |
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