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Mirrors > Home > MPE Home > Th. List > islmhm3 | Structured version Visualization version GIF version |
Description: Property of a module homomorphism, similar to ismhm 18749. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
islmhm.k | β’ πΎ = (Scalarβπ) |
islmhm.l | β’ πΏ = (Scalarβπ) |
islmhm.b | β’ π΅ = (BaseβπΎ) |
islmhm.e | β’ πΈ = (Baseβπ) |
islmhm.m | β’ Β· = ( Β·π βπ) |
islmhm.n | β’ Γ = ( Β·π βπ) |
Ref | Expression |
---|---|
islmhm3 | β’ ((π β LMod β§ π β LMod) β (πΉ β (π LMHom π) β (πΉ β (π GrpHom π) β§ πΏ = πΎ β§ βπ₯ β π΅ βπ¦ β πΈ (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmhm.k | . . 3 β’ πΎ = (Scalarβπ) | |
2 | islmhm.l | . . 3 β’ πΏ = (Scalarβπ) | |
3 | islmhm.b | . . 3 β’ π΅ = (BaseβπΎ) | |
4 | islmhm.e | . . 3 β’ πΈ = (Baseβπ) | |
5 | islmhm.m | . . 3 β’ Β· = ( Β·π βπ) | |
6 | islmhm.n | . . 3 β’ Γ = ( Β·π βπ) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 20919 | . 2 β’ (πΉ β (π LMHom π) β ((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ πΏ = πΎ β§ βπ₯ β π΅ βπ¦ β πΈ (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))))) |
8 | 7 | baib 534 | 1 β’ ((π β LMod β§ π β LMod) β (πΉ β (π LMHom π) β (πΉ β (π GrpHom π) β§ πΏ = πΎ β§ βπ₯ β π΅ βπ¦ β πΈ (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 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: islmhm2 20930 pj1lmhm 20992 |
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