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Mirrors > Home > MPE Home > Th. List > lmhmlem | Structured version Visualization version GIF version |
Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmlem.k | β’ πΎ = (Scalarβπ) |
lmhmlem.l | β’ πΏ = (Scalarβπ) |
Ref | Expression |
---|---|
lmhmlem | β’ (πΉ β (π LMHom π) β ((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ πΏ = πΎ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmlem.k | . . 3 β’ πΎ = (Scalarβπ) | |
2 | lmhmlem.l | . . 3 β’ πΏ = (Scalarβπ) | |
3 | eqid 2730 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | eqid 2730 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
5 | eqid 2730 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
6 | eqid 2730 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 20782 | . 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 (BaseβπΎ)βπ β (Baseβπ)(πΉβ(π( Β·π βπ)π)) = (π( Β·π βπ)(πΉβπ))))) |
8 | 3simpa 1146 | . . 3 β’ ((πΉ β (π GrpHom π) β§ πΏ = πΎ β§ βπ β (BaseβπΎ)βπ β (Baseβπ)(πΉβ(π( Β·π βπ)π)) = (π( Β·π βπ)(πΉβπ))) β (πΉ β (π GrpHom π) β§ πΏ = πΎ)) | |
9 | 8 | anim2i 615 | . 2 β’ (((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ πΏ = πΎ β§ βπ β (BaseβπΎ)βπ β (Baseβπ)(πΉβ(π( Β·π βπ)π)) = (π( Β·π βπ)(πΉβπ)))) β ((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ πΏ = πΎ))) |
10 | 7, 9 | sylbi 216 | 1 β’ (πΉ β (π LMHom π) β ((π β LMod β§ π β LMod) β§ (πΉ β (π GrpHom π) β§ πΏ = πΎ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 βwral 3059 βcfv 6542 (class class class)co 7411 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 GrpHom cghm 19127 LModclmod 20614 LMHom clmhm 20774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-lmhm 20777 |
This theorem is referenced by: lmhmsca 20785 lmghm 20786 lmhmlmod2 20787 lmhmlmod1 20788 |
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