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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 3200 | . . 3 β’ (π β βπ₯ β π βπ¦ β π (πΉβ(π₯ Β· π¦)) = (π₯ Γ (πΉβπ¦))) |
7 | 3, 4, 6 | 3jca 1128 | . 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 20637 | . 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 1344 | 1 β’ (π β πΉ β (π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 βcfv 6543 (class class class)co 7408 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 GrpHom cghm 19088 LModclmod 20470 LMHom clmhm 20629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-lmhm 20632 |
This theorem is referenced by: 0lmhm 20650 idlmhm 20651 invlmhm 20652 lmhmco 20653 lmhmplusg 20654 lmhmvsca 20655 lmhmf1o 20656 reslmhm2 20663 reslmhm2b 20664 pwsdiaglmhm 20667 pwssplit3 20671 frlmup1 21352 quslmhm 32465 lmhmqusker 32529 frlmsnic 41112 |
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