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Mirrors > Home > MPE Home > Th. List > lmhmrnlss | Structured version Visualization version GIF version |
Description: The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmrnlss | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2728 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | lmhmf 20913 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | ffn 6717 | . . 3 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆)) | |
5 | fnima 6680 | . . 3 ⊢ (𝐹 Fn (Base‘𝑆) → (𝐹 “ (Base‘𝑆)) = ran 𝐹) | |
6 | 3, 4, 5 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ (Base‘𝑆)) = ran 𝐹) |
7 | lmhmlmod1 20912 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) | |
8 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆) | |
9 | 1, 8 | lss1 20816 | . . . 4 ⊢ (𝑆 ∈ LMod → (Base‘𝑆) ∈ (LSubSp‘𝑆)) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘𝑆) ∈ (LSubSp‘𝑆)) |
11 | eqid 2728 | . . . 4 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇) | |
12 | 8, 11 | lmhmima 20926 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (Base‘𝑆) ∈ (LSubSp‘𝑆)) → (𝐹 “ (Base‘𝑆)) ∈ (LSubSp‘𝑇)) |
13 | 10, 12 | mpdan 686 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ (Base‘𝑆)) ∈ (LSubSp‘𝑇)) |
14 | 6, 13 | eqeltrrd 2830 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ran crn 5674 “ cima 5676 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 LModclmod 20737 LSubSpclss 20809 LMHom clmhm 20898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-ghm 19162 df-mgp 20069 df-ur 20116 df-ring 20169 df-lmod 20739 df-lss 20810 df-lmhm 20901 |
This theorem is referenced by: imlmhm 33310 dimkerim 33316 lmhmfgsplit 42501 lmhmlnmsplit 42502 |
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