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Theorem idlmhm 20931
Description: The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Hypothesis
Ref Expression
idlmhm.b 𝐡 = (Baseβ€˜π‘€)
Assertion
Ref Expression
idlmhm (𝑀 ∈ LMod β†’ ( I β†Ύ 𝐡) ∈ (𝑀 LMHom 𝑀))

Proof of Theorem idlmhm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idlmhm.b . 2 𝐡 = (Baseβ€˜π‘€)
2 eqid 2727 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2727 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
4 eqid 2727 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
5 id 22 . 2 (𝑀 ∈ LMod β†’ 𝑀 ∈ LMod)
6 eqidd 2728 . 2 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€))
7 lmodgrp 20755 . . 3 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
81idghm 19190 . . 3 (𝑀 ∈ Grp β†’ ( I β†Ύ 𝐡) ∈ (𝑀 GrpHom 𝑀))
97, 8syl 17 . 2 (𝑀 ∈ LMod β†’ ( I β†Ύ 𝐡) ∈ (𝑀 GrpHom 𝑀))
101, 3, 2, 4lmodvscl 20766 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝐡)
11103expb 1117 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝐡)
12 fvresi 7186 . . . 4 ((π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘€)𝑦))
1311, 12syl 17 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ 𝐡)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘€)𝑦))
14 fvresi 7186 . . . . 5 (𝑦 ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘¦) = 𝑦)
1514ad2antll 727 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ 𝐡)β€˜π‘¦) = 𝑦)
1615oveq2d 7440 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜π‘€)(( I β†Ύ 𝐡)β€˜π‘¦)) = (π‘₯( ·𝑠 β€˜π‘€)𝑦))
1713, 16eqtr4d 2770 . 2 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ 𝐡)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘€)(( I β†Ύ 𝐡)β€˜π‘¦)))
181, 2, 2, 3, 3, 4, 5, 5, 6, 9, 17islmhmd 20929 1 (𝑀 ∈ LMod β†’ ( I β†Ύ 𝐡) ∈ (𝑀 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   I cid 5577   β†Ύ cres 5682  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  Scalarcsca 17241   ·𝑠 cvsca 17242  Grpcgrp 18895   GrpHom cghm 19172  LModclmod 20748   LMHom clmhm 20909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-ghm 19173  df-lmod 20750  df-lmhm 20912
This theorem is referenced by:  idnmhm  24689  mendring  42619
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