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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 3194 | . . 3 β’ (π β βπ₯ β π βπ¦ β π (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))) |
7 | 3, 4, 6 | 3jca 1129 | . 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 20503 | . 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 1345 | 1 β’ (π β πΉ β (π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 GrpHom cghm 19010 LModclmod 20336 LMHom clmhm 20495 |
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-sep 5257 ax-nul 5264 ax-pr 5385 |
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-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-lmhm 20498 |
This theorem is referenced by: 0lmhm 20516 idlmhm 20517 invlmhm 20518 lmhmco 20519 lmhmplusg 20520 lmhmvsca 20521 lmhmf1o 20522 reslmhm2 20529 reslmhm2b 20530 pwsdiaglmhm 20533 pwssplit3 20537 frlmup1 21220 quslmhm 32194 frlmsnic 40771 |
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