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Theorem mendval 42439
Description: Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mendval.b 𝐵 = (𝑀 LMHom 𝑀)
mendval.p + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦))
mendval.t × = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
mendval.s 𝑆 = (Scalar‘𝑀)
mendval.v · = (𝑥 ∈ (Base‘𝑆), 𝑦𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
Assertion
Ref Expression
mendval (𝑀𝑋 → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑀,𝑦
Allowed substitution hints:   + (𝑥,𝑦)   𝑆(𝑥,𝑦)   · (𝑥,𝑦)   × (𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem mendval
Dummy variables 𝑚 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3485 . 2 (𝑀𝑋𝑀 ∈ V)
2 oveq12 7411 . . . . . . 7 ((𝑚 = 𝑀𝑚 = 𝑀) → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
32anidms 566 . . . . . 6 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
4 mendval.b . . . . . 6 𝐵 = (𝑀 LMHom 𝑀)
53, 4eqtr4di 2782 . . . . 5 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = 𝐵)
65csbeq1d 3890 . . . 4 (𝑚 = 𝑀(𝑚 LMHom 𝑚) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}) = 𝐵 / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}))
7 ovex 7435 . . . . . 6 (𝑚 LMHom 𝑚) ∈ V
85, 7eqeltrrdi 2834 . . . . 5 (𝑚 = 𝑀𝐵 ∈ V)
9 simpr 484 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑏 = 𝐵)
109opeq2d 4873 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 fveq2 6882 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
1211ofeqd 7666 . . . . . . . . . . 11 (𝑚 = 𝑀 → ∘f (+g𝑚) = ∘f (+g𝑀))
1312oveqdr 7430 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥f (+g𝑚)𝑦) = (𝑥f (+g𝑀)𝑦))
149, 9, 13mpoeq123dv 7477 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦)))
15 mendval.p . . . . . . . . 9 + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦))
1614, 15eqtr4di 2782 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = + )
1716opeq2d 4873 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩ = ⟨(+g‘ndx), + ⟩)
18 eqidd 2725 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑦) = (𝑥𝑦))
199, 9, 18mpoeq123dv 7477 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦)))
20 mendval.t . . . . . . . . 9 × = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
2119, 20eqtr4di 2782 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = × )
2221opeq2d 4873 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩ = ⟨(.r‘ndx), × ⟩)
2310, 17, 22tpeq123d 4745 . . . . . 6 ((𝑚 = 𝑀𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩})
24 fveq2 6882 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
2524adantr 480 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = (Scalar‘𝑀))
26 mendval.s . . . . . . . . 9 𝑆 = (Scalar‘𝑀)
2725, 26eqtr4di 2782 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = 𝑆)
2827opeq2d 4873 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Scalar‘ndx), (Scalar‘𝑚)⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
2927fveq2d 6886 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘(Scalar‘𝑚)) = (Base‘𝑆))
30 fveq2 6882 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
3130adantr 480 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → ( ·𝑠𝑚) = ( ·𝑠𝑀))
3231ofeqd 7666 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ∘f ( ·𝑠𝑚) = ∘f ( ·𝑠𝑀))
33 fveq2 6882 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3433adantr 480 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘𝑚) = (Base‘𝑀))
3534xpeq1d 5696 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ((Base‘𝑚) × {𝑥}) = ((Base‘𝑀) × {𝑥}))
36 eqidd 2725 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑦 = 𝑦)
3732, 35, 36oveq123d 7423 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
3829, 9, 37mpoeq123dv 7477 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦)) = (𝑥 ∈ (Base‘𝑆), 𝑦𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦)))
39 mendval.v . . . . . . . . 9 · = (𝑥 ∈ (Base‘𝑆), 𝑦𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
4038, 39eqtr4di 2782 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦)) = · )
4140opeq2d 4873 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
4228, 41preq12d 4738 . . . . . 6 ((𝑚 = 𝑀𝑏 = 𝐵) → {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩} = {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})
4323, 42uneq12d 4157 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
448, 43csbied 3924 . . . 4 (𝑚 = 𝑀𝐵 / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
456, 44eqtrd 2764 . . 3 (𝑚 = 𝑀(𝑚 LMHom 𝑚) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
46 df-mend 42432 . . 3 MEndo = (𝑚 ∈ V ↦ (𝑚 LMHom 𝑚) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩}))
47 tpex 7728 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
48 prex 5423 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩} ∈ V
4947, 48unex 7727 . . 3 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) ∈ V
5045, 46, 49fvmpt 6989 . 2 (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
511, 50syl 17 1 (𝑀𝑋 → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  csb 3886  cun 3939  {csn 4621  {cpr 4623  {ctp 4625  cop 4627   × cxp 5665  ccom 5671  cfv 6534  (class class class)co 7402  cmpo 7404  f cof 7662  ndxcnx 17127  Basecbs 17145  +gcplusg 17198  .rcmulr 17199  Scalarcsca 17201   ·𝑠 cvsca 17202   LMHom clmhm 20859  MEndocmend 42431
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-mend 42432
This theorem is referenced by:  mendbas  42440  mendplusgfval  42441  mendmulrfval  42443  mendsca  42445  mendvscafval  42446
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