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Mirrors > Home > MPE Home > Th. List > islmhmd | Structured version Visualization version GIF version |
Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
Ref | Expression |
---|---|
islmhmd.x | β’ π = (Baseβπ) |
islmhmd.a | β’ Β· = ( Β·π βπ) |
islmhmd.b | β’ Γ = ( Β·π βπ) |
islmhmd.k | β’ πΎ = (Scalarβπ) |
islmhmd.j | β’ π½ = (Scalarβπ) |
islmhmd.n | β’ π = (BaseβπΎ) |
islmhmd.s | β’ (π β π β LMod) |
islmhmd.t | β’ (π β π β LMod) |
islmhmd.c | β’ (π β π½ = πΎ) |
islmhmd.f | β’ (π β πΉ β (π GrpHom π)) |
islmhmd.l | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))) |
Ref | Expression |
---|---|
islmhmd | β’ (π β πΉ β (π LMHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmhmd.s | . 2 β’ (π β π β LMod) | |
2 | islmhmd.t | . 2 β’ (π β π β LMod) | |
3 | islmhmd.f | . . 3 β’ (π β πΉ β (π GrpHom π)) | |
4 | islmhmd.c | . . 3 β’ (π β π½ = πΎ) | |
5 | islmhmd.l | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))) | |
6 | 5 | ralrimivva 3198 | . . 3 β’ (π β βπ₯ β π βπ¦ β π (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))) |
7 | 3, 4, 6 | 3jca 1125 | . 2 β’ (π β (πΉ β (π GrpHom π) β§ π½ = πΎ β§ βπ₯ β π βπ¦ β π (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦)))) |
8 | islmhmd.k | . . 3 β’ πΎ = (Scalarβπ) | |
9 | islmhmd.j | . . 3 β’ π½ = (Scalarβπ) | |
10 | islmhmd.n | . . 3 β’ π = (BaseβπΎ) | |
11 | islmhmd.x | . . 3 β’ π = (Baseβπ) | |
12 | islmhmd.a | . . 3 β’ Β· = ( Β·π βπ) | |
13 | islmhmd.b | . . 3 β’ Γ = ( Β·π βπ) | |
14 | 8, 9, 10, 11, 12, 13 | islmhm 20919 | . 2 β’ (πΉ β (π LMHom π) β ((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ π½ = πΎ β§ βπ₯ β π βπ¦ β π (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))))) |
15 | 1, 2, 7, 14 | syl21anbrc 1341 | 1 β’ (π β πΉ β (π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 βcfv 6553 (class class class)co 7426 Basecbs 17187 Scalarcsca 17243 Β·π cvsca 17244 GrpHom cghm 19174 LModclmod 20750 LMHom clmhm 20911 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-lmhm 20914 |
This theorem is referenced by: 0lmhm 20932 idlmhm 20933 invlmhm 20934 lmhmco 20935 lmhmplusg 20936 lmhmvsca 20937 lmhmf1o 20938 reslmhm2 20945 reslmhm2b 20946 pwsdiaglmhm 20949 pwssplit3 20953 frlmup1 21739 imaslmhm 33093 quslmhm 33095 lmhmqusker 33152 frlmsnic 41801 |
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