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Theorem mendsca 37226
Description: The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mendsca.a 𝐴 = (MEndo‘𝑀)
mendsca.s 𝑆 = (Scalar‘𝑀)
Assertion
Ref Expression
mendsca 𝑆 = (Scalar‘𝐴)

Proof of Theorem mendsca
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6160 . . . . 5 (Scalar‘𝑀) ∈ V
2 eqid 2626 . . . . . 6 ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩}) = ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩})
32algsca 37218 . . . . 5 ((Scalar‘𝑀) ∈ V → (Scalar‘𝑀) = (Scalar‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩})))
41, 3mp1i 13 . . . 4 (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩})))
5 eqid 2626 . . . . . 6 (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
6 eqid 2626 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))
7 eqid 2626 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))
8 eqid 2626 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
9 eqid 2626 . . . . . 6 (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦)) = (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))
105, 6, 7, 8, 9mendval 37220 . . . . 5 (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩}))
1110fveq2d 6154 . . . 4 (𝑀 ∈ V → (Scalar‘(MEndo‘𝑀)) = (Scalar‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑓 (+g𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))⟩})))
124, 11eqtr4d 2663 . . 3 (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
13 df-sca 15873 . . . . 5 Scalar = Slot 5
1413str0 15827 . . . 4 ∅ = (Scalar‘∅)
15 fvprc 6144 . . . 4 𝑀 ∈ V → (Scalar‘𝑀) = ∅)
16 fvprc 6144 . . . . 5 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
1716fveq2d 6154 . . . 4 𝑀 ∈ V → (Scalar‘(MEndo‘𝑀)) = (Scalar‘∅))
1814, 15, 173eqtr4a 2686 . . 3 𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
1912, 18pm2.61i 176 . 2 (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀))
20 mendsca.s . 2 𝑆 = (Scalar‘𝑀)
21 mendsca.a . . 3 𝐴 = (MEndo‘𝑀)
2221fveq2i 6153 . 2 (Scalar‘𝐴) = (Scalar‘(MEndo‘𝑀))
2319, 20, 223eqtr4i 2658 1 𝑆 = (Scalar‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  c0 3896  {csn 4153  {cpr 4155  {ctp 4157  cop 4159   × cxp 5077  ccom 5083  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑓 cof 6849  5c5 11018  ndxcnx 15773  Basecbs 15776  +gcplusg 15857  .rcmulr 15858  Scalarcsca 15860   ·𝑠 cvsca 15861   LMHom clmhm 18933  MEndocmend 37212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-mend 37213
This theorem is referenced by:  mendlmod  37230  mendassa  37231
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