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Theorem mendval 39768
 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 3511 . 2 (𝑀𝑋𝑀 ∈ V)
2 oveq12 7157 . . . . . . 7 ((𝑚 = 𝑀𝑚 = 𝑀) → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
32anidms 569 . . . . . 6 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = (𝑀 LMHom 𝑀))
4 mendval.b . . . . . 6 𝐵 = (𝑀 LMHom 𝑀)
53, 4syl6eqr 2872 . . . . 5 (𝑚 = 𝑀 → (𝑚 LMHom 𝑚) = 𝐵)
65csbeq1d 3885 . . . 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 7181 . . . . . 6 (𝑚 LMHom 𝑚) ∈ V
85, 7eqeltrrdi 2920 . . . . 5 (𝑚 = 𝑀𝐵 ∈ V)
9 simpr 487 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑏 = 𝐵)
109opeq2d 4802 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 fveq2 6663 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
12 ofeq 7403 . . . . . . . . . . . 12 ((+g𝑚) = (+g𝑀) → ∘f (+g𝑚) = ∘f (+g𝑀))
1311, 12syl 17 . . . . . . . . . . 11 (𝑚 = 𝑀 → ∘f (+g𝑚) = ∘f (+g𝑀))
1413oveqdr 7176 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥f (+g𝑚)𝑦) = (𝑥f (+g𝑀)𝑦))
159, 9, 14mpoeq123dv 7221 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦)))
16 mendval.p . . . . . . . . 9 + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f (+g𝑀)𝑦))
1715, 16syl6eqr 2872 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦)) = + )
1817opeq2d 4802 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥f (+g𝑚)𝑦))⟩ = ⟨(+g‘ndx), + ⟩)
19 eqidd 2820 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑦) = (𝑥𝑦))
209, 9, 19mpoeq123dv 7221 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦)))
21 mendval.t . . . . . . . . 9 × = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
2220, 21syl6eqr 2872 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦)) = × )
2322opeq2d 4802 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩ = ⟨(.r‘ndx), × ⟩)
2410, 18, 23tpeq123d 4676 . . . . . 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 6663 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
2625adantr 483 . . . . . . . . 9 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = (Scalar‘𝑀))
27 mendval.s . . . . . . . . 9 𝑆 = (Scalar‘𝑀)
2826, 27syl6eqr 2872 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (Scalar‘𝑚) = 𝑆)
2928opeq2d 4802 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨(Scalar‘ndx), (Scalar‘𝑚)⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
3028fveq2d 6667 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘(Scalar‘𝑚)) = (Base‘𝑆))
31 fveq2 6663 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
3231adantr 483 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → ( ·𝑠𝑚) = ( ·𝑠𝑀))
33 ofeq 7403 . . . . . . . . . . . 12 (( ·𝑠𝑚) = ( ·𝑠𝑀) → ∘f ( ·𝑠𝑚) = ∘f ( ·𝑠𝑀))
3432, 33syl 17 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ∘f ( ·𝑠𝑚) = ∘f ( ·𝑠𝑀))
35 fveq2 6663 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3635adantr 483 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑏 = 𝐵) → (Base‘𝑚) = (Base‘𝑀))
3736xpeq1d 5577 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → ((Base‘𝑚) × {𝑥}) = ((Base‘𝑀) × {𝑥}))
38 eqidd 2820 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑏 = 𝐵) → 𝑦 = 𝑦)
3934, 37, 38oveq123d 7169 . . . . . . . . . 10 ((𝑚 = 𝑀𝑏 = 𝐵) → (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
4030, 9, 39mpoeq123dv 7221 . . . . . . . . 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, 41syl6eqr 2872 . . . . . . . 8 ((𝑚 = 𝑀𝑏 = 𝐵) → (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦)) = · )
4342opeq2d 4802 . . . . . . 7 ((𝑚 = 𝑀𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠𝑚)𝑦))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
4429, 43preq12d 4669 . . . . . 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 4138 . . . . 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 3917 . . . 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 2854 . . 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 39761 . . 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 7462 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
50 prex 5323 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩} ∈ V
5149, 50unex 7461 . . 3 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) ∈ V
5247, 48, 51fvmpt 6761 . 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 398   = wceq 1530   ∈ wcel 2107  Vcvv 3493  ⦋csb 3881   ∪ cun 3932  {csn 4559  {cpr 4561  {ctp 4563  ⟨cop 4565   × cxp 5546   ∘ ccom 5552  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150   ∘f cof 7399  ndxcnx 16472  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561   LMHom clmhm 19783  MEndocmend 39760 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-mend 39761 This theorem is referenced by:  mendbas  39769  mendplusgfval  39770  mendmulrfval  39772  mendsca  39774  mendvscafval  39775
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