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Mirrors > Home > MPE Home > Th. List > idlmhm | Structured version Visualization version GIF version |
Description: The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
idlmhm.b | ⊢ 𝐵 = (Base‘𝑀) |
Ref | Expression |
---|---|
idlmhm | ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlmhm.b | . 2 ⊢ 𝐵 = (Base‘𝑀) | |
2 | eqid 2821 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2821 | . 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
4 | eqid 2821 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
5 | id 22 | . 2 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod) | |
6 | eqidd 2822 | . 2 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀)) | |
7 | lmodgrp 19635 | . . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
8 | 1 | idghm 18367 | . . 3 ⊢ (𝑀 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀)) |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀)) |
10 | 1, 3, 2, 4 | lmodvscl 19645 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
11 | 10 | 3expb 1116 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
12 | fvresi 6929 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦)) |
14 | fvresi 6929 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
15 | 14 | ad2antll 727 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦) |
16 | 15 | oveq2d 7166 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦)) |
17 | 13, 16 | eqtr4d 2859 | . 2 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦))) |
18 | 1, 2, 2, 3, 3, 4, 5, 5, 6, 9, 17 | islmhmd 19805 | 1 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 I cid 5453 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 Grpcgrp 18097 GrpHom cghm 18349 LModclmod 19628 LMHom clmhm 19785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-ghm 18350 df-lmod 19630 df-lmhm 19788 |
This theorem is referenced by: idnmhm 23357 mendring 39785 |
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