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Theorem mendval 40740
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 3439 . 2 (𝑀𝑋𝑀 ∈ V)
2 oveq12 7241 . . . . . . 7 ((𝑚 = 𝑀𝑚 = 𝑀) → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
32anidms 570 . . . . . 6 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
4 mendval.b . . . . . 6 𝐵 = (𝑀 LMHom 𝑀)
53, 4eqtr4di 2797 . . . . 5 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = 𝐵)
65csbeq1d 3830 . . . 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 7265 . . . . . 6 (𝑚 LMHom 𝑚) ∈ V
85, 7eqeltrrdi 2848 . . . . 5 (𝑚 = 𝑀𝐵 ∈ V)
9 simpr 488 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑏 = 𝐵)
109opeq2d 4806 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 fveq2 6736 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
12 ofeq 7490 . . . . . . . . . . . 12 ((+g𝑚) = (+g𝑀) → ∘f (+g𝑚) = ∘f (+g𝑀))
1311, 12syl 17 . . . . . . . . . . 11 (𝑚 = 𝑀 → ∘f (+g𝑚) = ∘f (+g𝑀))
1413oveqdr 7260 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥f (+g𝑚)𝑦) = (𝑥f (+g𝑀)𝑦))
159, 9, 14mpoeq123dv 7305 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦)))
16 mendval.p . . . . . . . . 9 + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦))
1715, 16eqtr4di 2797 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = + )
1817opeq2d 4806 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩ = ⟨(+g‘ndx), + ⟩)
19 eqidd 2739 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑦) = (𝑥𝑦))
209, 9, 19mpoeq123dv 7305 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦)))
21 mendval.t . . . . . . . . 9 × = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
2220, 21eqtr4di 2797 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = × )
2322opeq2d 4806 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩ = ⟨(.r‘ndx), × ⟩)
2410, 18, 23tpeq123d 4679 . . . . . 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), × ⟩})
25 fveq2 6736 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
2625adantr 484 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = (Scalar‘𝑀))
27 mendval.s . . . . . . . . 9 𝑆 = (Scalar‘𝑀)
2826, 27eqtr4di 2797 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = 𝑆)
2928opeq2d 4806 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Scalar‘ndx), (Scalar‘𝑚)⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
3028fveq2d 6740 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘(Scalar‘𝑚)) = (Base‘𝑆))
31 fveq2 6736 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
3231adantr 484 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → ( ·𝑠𝑚) = ( ·𝑠𝑀))
33 ofeq 7490 . . . . . . . . . . . 12 (( ·𝑠𝑚) = ( ·𝑠𝑀) → ∘f ( ·𝑠𝑚) = ∘f ( ·𝑠𝑀))
3432, 33syl 17 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ∘f ( ·𝑠𝑚) = ∘f ( ·𝑠𝑀))
35 fveq2 6736 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3635adantr 484 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘𝑚) = (Base‘𝑀))
3736xpeq1d 5595 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ((Base‘𝑚) × {𝑥}) = ((Base‘𝑀) × {𝑥}))
38 eqidd 2739 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑦 = 𝑦)
3934, 37, 38oveq123d 7253 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
4030, 9, 39mpoeq123dv 7305 . . . . . . . . 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)))
41 mendval.v . . . . . . . . 9 · = (𝑥 ∈ (Base‘𝑆), 𝑦𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
4240, 41eqtr4di 2797 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦)) = · )
4342opeq2d 4806 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
4429, 43preq12d 4672 . . . . . 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), · ⟩})
4524, 44uneq12d 4093 . . . . 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), · ⟩}))
468, 45csbied 3864 . . . 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), · ⟩}))
476, 46eqtrd 2778 . . 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), · ⟩}))
48 df-mend 40733 . . 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))⟩}))
49 tpex 7551 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
50 prex 5340 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩} ∈ V
5149, 50unex 7550 . . 3 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) ∈ V
5247, 48, 51fvmpt 6837 . 2 (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
531, 52syl 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 399   = wceq 1543  wcel 2111  Vcvv 3421  csb 3826  cun 3879  {csn 4556  {cpr 4558  {ctp 4560  cop 4562   × cxp 5564  ccom 5570  cfv 6398  (class class class)co 7232  cmpo 7234  f cof 7486  ndxcnx 16769  Basecbs 16785  +gcplusg 16827  .rcmulr 16828  Scalarcsca 16830   ·𝑠 cvsca 16831   LMHom clmhm 20081  MEndocmend 40732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-iota 6356  df-fun 6400  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-of 7488  df-mend 40733
This theorem is referenced by:  mendbas  40741  mendplusgfval  40742  mendmulrfval  40744  mendsca  40746  mendvscafval  40747
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