Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mendsca Structured version   Visualization version   GIF version

Theorem mendsca 42886
Description: The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
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 6905 . . . . 5 (Scalar‘𝑀) ∈ V
2 eqid 2726 . . . . . 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 42878 . . . . 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 2726 . . . . . 6 (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
6 eqid 2726 . . . . . 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 2726 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥𝑦))
8 eqid 2726 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
9 eqid 2726 . . . . . 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 42880 . . . . 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 6896 . . . 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 2769 . . 3 (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
13 scaid 17323 . . . . . 6 Scalar = Slot (Scalar‘ndx)
1413str0 17185 . . . . 5 ∅ = (Scalar‘∅)
1514eqcomi 2735 . . . 4 (Scalar‘∅) = ∅
16 eqid 2726 . . . 4 (MEndo‘𝑀) = (MEndo‘𝑀)
1715, 16fveqprc 17187 . . 3 𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
1812, 17pm2.61i 182 . 2 (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀))
19 mendsca.s . 2 𝑆 = (Scalar‘𝑀)
20 mendsca.a . . 3 𝐴 = (MEndo‘𝑀)
2120fveq2i 6895 . 2 (Scalar‘𝐴) = (Scalar‘(MEndo‘𝑀))
2218, 19, 213eqtr4i 2764 1 𝑆 = (Scalar‘𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3464  cun 3946  c0 4324  {csn 4625  {cpr 4627  {ctp 4629  cop 4631   × cxp 5672  ccom 5678  cfv 6545  (class class class)co 7415  cmpo 7417  f cof 7679  ndxcnx 17189  Basecbs 17207  +gcplusg 17260  .rcmulr 17261  Scalarcsca 17263   ·𝑠 cvsca 17264   LMHom clmhm 20992  MEndocmend 42872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8966  df-dom 8967  df-sdom 8968  df-fin 8969  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12258  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-n0 12518  df-z 12604  df-uz 12868  df-fz 13532  df-struct 17143  df-slot 17178  df-ndx 17190  df-base 17208  df-plusg 17273  df-mulr 17274  df-sca 17276  df-vsca 17277  df-mend 42873
This theorem is referenced by:  mendlmod  42890  mendassa  42891
  Copyright terms: Public domain W3C validator