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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrlss Structured version   Visualization version   GIF version
Theorem lkrlss 38013
Description: The kernel of a linear functional is a subspace. (nlelshi 31344 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrlss.f 𝐹 = (LFnlβ€˜π‘Š)
lkrlss.k 𝐾 = (LKerβ€˜π‘Š)
lkrlss.s 𝑆 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
lkrlss ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) ∈ 𝑆)
Proof of Theorem lkrlss
Dummy variables π‘₯ π‘Ÿ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2733 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
4 lkrlss.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
5 lkrlss.k . . . 4 𝐾 = (LKerβ€˜π‘Š)
61, 2, 3, 4, 5lkrval2 38008 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) = {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))})
7 ssrab2 4078 . . 3 {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))} βŠ† (Baseβ€˜π‘Š)
86, 7eqsstrdi 4037 . 2 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) βŠ† (Baseβ€˜π‘Š))
9 eqid 2733 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
101, 9lmod0vcl 20501 . . . . 5 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
1110adantr 482 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (0gβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
122, 3, 9, 4lfl0 37983 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΊβ€˜(0gβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š)))
131, 2, 3, 4, 5ellkr 38007 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ ((0gβ€˜π‘Š) ∈ (πΎβ€˜πΊ) ↔ ((0gβ€˜π‘Š) ∈ (Baseβ€˜π‘Š) ∧ (πΊβ€˜(0gβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š)))))
1411, 12, 13mpbir2and 712 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (0gβ€˜π‘Š) ∈ (πΎβ€˜πΊ))
1514ne0d 4336 . 2 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) β‰  βˆ…)
16 simplll 774 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ π‘Š ∈ LMod)
17 simplr 768 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
18 simpllr 775 . . . . . . . 8 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ 𝐺 ∈ 𝐹)
19 simprl 770 . . . . . . . 8 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ π‘₯ ∈ (πΎβ€˜πΊ))
201, 4, 5lkrcl 38010 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ π‘₯ ∈ (πΎβ€˜πΊ)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
2116, 18, 19, 20syl3anc 1372 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
22 eqid 2733 . . . . . . . 8 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
23 eqid 2733 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
241, 2, 22, 23lmodvscl 20489 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š))
2516, 17, 21, 24syl3anc 1372 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š))
26 simprr 772 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ 𝑦 ∈ (πΎβ€˜πΊ))
271, 4, 5lkrcl 38010 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (πΎβ€˜πΊ)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
2816, 18, 26, 27syl3anc 1372 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
29 eqid 2733 . . . . . . 7 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
301, 29lmodvacl 20486 . . . . . 6 ((π‘Š ∈ LMod ∧ (π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
3116, 25, 28, 30syl3anc 1372 . . . . 5 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
32 eqid 2733 . . . . . . . 8 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
33 eqid 2733 . . . . . . . 8 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
341, 29, 2, 22, 23, 32, 33, 4lfli 37979 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))
3516, 18, 17, 21, 28, 34syl113anc 1383 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))
362, 3, 4, 5lkrf0 38011 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ π‘₯ ∈ (πΎβ€˜πΊ)) β†’ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
3716, 18, 19, 36syl3anc 1372 . . . . . . . . 9 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
3837oveq2d 7425 . . . . . . . 8 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯)) = (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))))
392lmodring 20479 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Ring)
4016, 39syl 17 . . . . . . . . 9 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (Scalarβ€˜π‘Š) ∈ Ring)
4123, 33, 3ringrz 20108 . . . . . . . . 9 (((Scalarβ€˜π‘Š) ∈ Ring ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
4240, 17, 41syl2anc 585 . . . . . . . 8 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
4338, 42eqtrd 2773 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯)) = (0gβ€˜(Scalarβ€˜π‘Š)))
442, 3, 4, 5lkrf0 38011 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (πΎβ€˜πΊ)) β†’ (πΊβ€˜π‘¦) = (0gβ€˜(Scalarβ€˜π‘Š)))
4516, 18, 26, 44syl3anc 1372 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (πΊβ€˜π‘¦) = (0gβ€˜(Scalarβ€˜π‘Š)))
4643, 45oveq12d 7427 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)) = ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))))
472lmodfgrp 20480 . . . . . . . 8 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
4816, 47syl 17 . . . . . . 7 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (Scalarβ€˜π‘Š) ∈ Grp)
4923, 3grpidcl 18850 . . . . . . 7 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
5023, 32, 3grplid 18852 . . . . . . 7 (((Scalarβ€˜π‘Š) ∈ Grp ∧ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5148, 49, 50syl2anc2 586 . . . . . 6 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5235, 46, 513eqtrd 2777 . . . . 5 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = (0gβ€˜(Scalarβ€˜π‘Š)))
531, 2, 3, 4, 5ellkr 38007 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ) ↔ (((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š) ∧ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = (0gβ€˜(Scalarβ€˜π‘Š)))))
5453ad2antrr 725 . . . . 5 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ) ↔ (((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š) ∧ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = (0gβ€˜(Scalarβ€˜π‘Š)))))
5531, 52, 54mpbir2and 712 . . . 4 ((((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘₯ ∈ (πΎβ€˜πΊ) ∧ 𝑦 ∈ (πΎβ€˜πΊ))) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ))
5655ralrimivva 3201 . . 3 (((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ βˆ€π‘₯ ∈ (πΎβ€˜πΊ)βˆ€π‘¦ ∈ (πΎβ€˜πΊ)((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ))
5756ralrimiva 3147 . 2 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (πΎβ€˜πΊ)βˆ€π‘¦ ∈ (πΎβ€˜πΊ)((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ))
58 lkrlss.s . . 3 𝑆 = (LSubSpβ€˜π‘Š)
592, 23, 1, 29, 22, 58islss 20545 . 2 ((πΎβ€˜πΊ) ∈ 𝑆 ↔ ((πΎβ€˜πΊ) βŠ† (Baseβ€˜π‘Š) ∧ (πΎβ€˜πΊ) β‰  βˆ… ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (πΎβ€˜πΊ)βˆ€π‘¦ ∈ (πΎβ€˜πΊ)((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦) ∈ (πΎβ€˜πΊ)))
608, 15, 57, 59syl3anbrc 1344 1 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433   βŠ† wss 3949  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  Grpcgrp 18819  Ringcrg 20056  LModclmod 20471  LSubSpclss 20542  LFnlclfn 37975  LKerclk 38003
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mgp 19988  df-ur 20005  df-ring 20058  df-lmod 20473  df-lss 20543  df-lfl 37976  df-lkr 38004
This theorem is referenced by:  lkrssv  38014  lkrlsp  38020  lkrlsp3  38022  lkrshp  38023  lclkrlem2f  40431  lclkrlem2n  40439  lclkrlem2v  40447  lcfrlem25  40486  lcfrlem35  40496
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