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Mirrors > Home > MPE Home > Th. List > lmhmkerlss | Structured version Visualization version GIF version |
Description: The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmkerlss.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
lmhmkerlss.z | ⊢ 0 = (0g‘𝑇) |
lmhmkerlss.u | ⊢ 𝑈 = (LSubSp‘𝑆) |
Ref | Expression |
---|---|
lmhmkerlss | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmkerlss.k | . 2 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
2 | lmhmlmod2 19538 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod) | |
3 | lmhmkerlss.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
4 | eqid 2771 | . . . . 5 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇) | |
5 | 3, 4 | lsssn0 19453 | . . . 4 ⊢ (𝑇 ∈ LMod → { 0 } ∈ (LSubSp‘𝑇)) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → { 0 } ∈ (LSubSp‘𝑇)) |
7 | lmhmkerlss.u | . . . 4 ⊢ 𝑈 = (LSubSp‘𝑆) | |
8 | 7, 4 | lmhmpreima 19554 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ { 0 } ∈ (LSubSp‘𝑇)) → (◡𝐹 “ { 0 }) ∈ 𝑈) |
9 | 6, 8 | mpdan 675 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (◡𝐹 “ { 0 }) ∈ 𝑈) |
10 | 1, 9 | syl5eqel 2863 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {csn 4435 ◡ccnv 5402 “ cima 5406 ‘cfv 6185 (class class class)co 6974 0gc0g 16567 LModclmod 19368 LSubSpclss 19437 LMHom clmhm 19525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-ghm 18139 df-mgp 18975 df-ur 18987 df-ring 19034 df-lmod 19370 df-lss 19438 df-lmhm 19528 |
This theorem is referenced by: frlmsslss 20635 kerlmhm 30679 dimkerim 30684 kercvrlsm 39117 lmhmfgsplit 39120 lmhmlnmsplit 39121 pwssplit4 39123 |
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