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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 2737 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
| 3 | eqid 2737 | . 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
| 4 | eqid 2737 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
| 5 | id 22 | . 2 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod) | |
| 6 | eqidd 2738 | . 2 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀)) | |
| 7 | lmodgrp 20865 | . . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
| 8 | 1 | idghm 19249 | . . 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 20876 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
| 11 | 10 | 3expb 1121 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
| 12 | fvresi 7193 | . . . 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 7193 | . . . . 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 7447 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦)) |
| 17 | 13, 16 | eqtr4d 2780 | . 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 21038 | 1 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 I cid 5577 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 Grpcgrp 18951 GrpHom cghm 19230 LModclmod 20858 LMHom clmhm 21018 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ghm 19231 df-lmod 20860 df-lmhm 21021 |
| This theorem is referenced by: idnmhm 24775 mendring 43200 |
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