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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmhmlvec2 | Structured version Visualization version GIF version |
Description: A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.) |
Ref | Expression |
---|---|
lmhmlvec2 | β’ ((π β LVec β§ πΉ β (π LMHom π)) β π β LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmlmod2 20921 | . . 3 β’ (πΉ β (π LMHom π) β π β LMod) | |
2 | 1 | adantl 480 | . 2 β’ ((π β LVec β§ πΉ β (π LMHom π)) β π β LMod) |
3 | eqid 2725 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2725 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
5 | 3, 4 | lmhmsca 20919 | . . . 4 β’ (πΉ β (π LMHom π) β (Scalarβπ) = (Scalarβπ)) |
6 | 5 | adantl 480 | . . 3 β’ ((π β LVec β§ πΉ β (π LMHom π)) β (Scalarβπ) = (Scalarβπ)) |
7 | 3 | lvecdrng 20994 | . . . 4 β’ (π β LVec β (Scalarβπ) β DivRing) |
8 | 7 | adantr 479 | . . 3 β’ ((π β LVec β§ πΉ β (π LMHom π)) β (Scalarβπ) β DivRing) |
9 | 6, 8 | eqeltrd 2825 | . 2 β’ ((π β LVec β§ πΉ β (π LMHom π)) β (Scalarβπ) β DivRing) |
10 | 4 | islvec 20993 | . 2 β’ (π β LVec β (π β LMod β§ (Scalarβπ) β DivRing)) |
11 | 2, 9, 10 | sylanbrc 581 | 1 β’ ((π β LVec β§ πΉ β (π LMHom π)) β π β LVec) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Scalarcsca 17235 DivRingcdr 20628 LModclmod 20747 LMHom clmhm 20908 LVecclvec 20991 |
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-sep 5294 ax-nul 5301 ax-pr 5423 |
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-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-lmhm 20911 df-lvec 20992 |
This theorem is referenced by: imlmhm 33376 dimkerim 33382 algextdeglem8 33449 |
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