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Theorem lkrlss 37603
Description: The kernel of a linear functional is a subspace. (nlelshi 31044 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 37598 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΎβ€˜πΊ) = {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))})
7 ssrab2 4038 . . 3 {π‘₯ ∈ (Baseβ€˜π‘Š) ∣ (πΊβ€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))} βŠ† (Baseβ€˜π‘Š)
86, 7eqsstrdi 3999 . 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 20366 . . . . 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 37573 . . . 4 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ (πΊβ€˜(0gβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š)))
131, 2, 3, 4, 5ellkr 37597 . . . 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 4296 . 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 37600 . . . . . . . 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 20354 . . . . . . 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 37600 . . . . . . 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 20351 . . . . . 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 37569 . . . . . . 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 37601 . . . . . . . . . 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 7374 . . . . . . . 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 20344 . . . . . . . . . 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 20017 . . . . . . . . 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 37601 . . . . . . . 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 7376 . . . . . 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 20345 . . . . . . . 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 18783 . . . . . . 7 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
5023, 32, 3grplid 18785 . . . . . . 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 37597 . . . . . 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 3194 . . 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 3140 . 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 20410 . 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 2940  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  βˆ…c0 4283  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  Grpcgrp 18753  Ringcrg 19969  LModclmod 20336  LSubSpclss 20407  LFnlclfn 37565  LKerclk 37593
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338  df-lss 20408  df-lfl 37566  df-lkr 37594
This theorem is referenced by:  lkrssv  37604  lkrlsp  37610  lkrlsp3  37612  lkrshp  37613  lclkrlem2f  40021  lclkrlem2n  40029  lclkrlem2v  40037  lcfrlem25  40076  lcfrlem35  40086
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