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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 2734 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2734 | . 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
4 | eqid 2734 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
5 | id 22 | . 2 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod) | |
6 | eqidd 2735 | . 2 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀)) | |
7 | lmodgrp 20881 | . . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
8 | 1 | idghm 19261 | . . 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 20892 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
11 | 10 | 3expb 1119 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
12 | fvresi 7192 | . . . 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 7192 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
15 | 14 | ad2antll 729 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦) |
16 | 15 | oveq2d 7446 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦)) |
17 | 13, 16 | eqtr4d 2777 | . 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 21055 | 1 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 I cid 5581 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Scalarcsca 17300 ·𝑠 cvsca 17301 Grpcgrp 18963 GrpHom cghm 19242 LModclmod 20874 LMHom clmhm 21035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ghm 19243 df-lmod 20876 df-lmhm 21038 |
This theorem is referenced by: idnmhm 24790 mendring 43176 |
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