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Theorem lkrlss 37953
Description: The kernel of a linear functional is a subspace. (nlelshi 31300 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 2732 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2732 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2732 . . . 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 37948 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) = {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))})
7 ssrab2 4076 . . 3 {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))} βŠ† (Baseβ€˜π‘Š)
86, 7eqsstrdi 4035 . 2 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) βŠ† (Baseβ€˜π‘Š))
9 eqid 2732 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
101, 9lmod0vcl 20493 . . . . 5 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
1110adantr 481 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (0gβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
122, 3, 9, 4lfl0 37923 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΊβ€˜(0gβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š)))
131, 2, 3, 4, 5ellkr 37947 . . . 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 711 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (0gβ€˜π‘Š) ∈ (πΎβ€˜πΊ))
1514ne0d 4334 . 2 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) β‰  βˆ…)
16 simplll 773 . . . . . 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 767 . . . . . . 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 774 . . . . . . . 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 769 . . . . . . . 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 37950 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ π‘₯ ∈ (πΎβ€˜πΊ)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
2116, 18, 19, 20syl3anc 1371 . . . . . . 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 2732 . . . . . . . 8 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
23 eqid 2732 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
241, 2, 22, 23lmodvscl 20481 . . . . . . 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 1371 . . . . . 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 771 . . . . . . 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 37950 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (πΎβ€˜πΊ)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
2816, 18, 26, 27syl3anc 1371 . . . . . 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 2732 . . . . . . 7 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
301, 29lmodvacl 20478 . . . . . 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 1371 . . . . 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 2732 . . . . . . . 8 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
33 eqid 2732 . . . . . . . 8 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
341, 29, 2, 22, 23, 32, 33, 4lfli 37919 . . . . . . 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 1382 . . . . . 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 37951 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ π‘₯ ∈ (πΎβ€˜πΊ)) β†’ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
3716, 18, 19, 36syl3anc 1371 . . . . . . . . 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 7421 . . . . . . . 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 20471 . . . . . . . . . 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 20101 . . . . . . . . 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 584 . . . . . . . 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 2772 . . . . . . 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 37951 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (πΎβ€˜πΊ)) β†’ (πΊβ€˜π‘¦) = (0gβ€˜(Scalarβ€˜π‘Š)))
4516, 18, 26, 44syl3anc 1371 . . . . . . 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 7423 . . . . . 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 20472 . . . . . . . 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 18846 . . . . . . 7 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
5023, 32, 3grplid 18848 . . . . . . 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 585 . . . . . 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 2776 . . . . 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 37947 . . . . . 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 724 . . . . 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 711 . . . 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 3200 . . 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 3146 . 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 20537 . 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 1343 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432   βŠ† wss 3947  βˆ…c0 4321  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  Grpcgrp 18815  Ringcrg 20049  LModclmod 20463  LSubSpclss 20534  LFnlclfn 37915  LKerclk 37943
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mgp 19982  df-ur 19999  df-ring 20051  df-lmod 20465  df-lss 20535  df-lfl 37916  df-lkr 37944
This theorem is referenced by:  lkrssv  37954  lkrlsp  37960  lkrlsp3  37962  lkrshp  37963  lclkrlem2f  40371  lclkrlem2n  40379  lclkrlem2v  40387  lcfrlem25  40426  lcfrlem35  40436
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