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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 2733 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
5 | eqid 2733 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
6 | eqid 2733 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 20503 | . 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 1149 | . . 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 618 | . 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 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: lmhmsca 20506 lmghm 20507 lmhmlmod2 20508 lmhmlmod1 20509 |
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