Theorem mendbas 40925

Description: Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Hypothesis

Ref	Expression
mendbas.a	⊢ 𝐴 = (MEndo‘𝑀)

Assertion

Ref	Expression
mendbas	⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)

Proof of Theorem mendbas

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7288	. . . 4 ⊢ (𝑀 LMHom 𝑀) ∈ V
2		eqid 2738	. . . . 5 ⊢ ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}) = ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})
3	2	algbase 40919	. . . 4 ⊢ ((𝑀 LMHom 𝑀) ∈ V → (𝑀 LMHom 𝑀) = (Base‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
4	1, 3	mp1i 13	. . 3 ⊢ (𝑀 ∈ V → (𝑀 LMHom 𝑀) = (Base‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
5		mendbas.a	. . . . 5 ⊢ 𝐴 = (MEndo‘𝑀)
6		eqid 2738	. . . . . 6 ⊢ (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
7		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))
8		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))
9		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
10		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)) = (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
11	6, 7, 8, 9, 10	mendval 40924	. . . . 5 ⊢ (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}))
12	5, 11	syl5eq 2791	. . . 4 ⊢ (𝑀 ∈ V → 𝐴 = ({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}))
13	12	fveq2d 6760	. . 3 ⊢ (𝑀 ∈ V → (Base‘𝐴) = (Base‘({⟨(Base‘ndx), (𝑀 LMHom 𝑀)⟩, ⟨(+g‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
14	4, 13	eqtr4d 2781	. 2 ⊢ (𝑀 ∈ V → (𝑀 LMHom 𝑀) = (Base‘𝐴))
15		base0 16845	. . 3 ⊢ ∅ = (Base‘∅)
16		reldmlmhm 20202	. . . 4 ⊢ Rel dom LMHom
17	16	ovprc1 7294	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑀 LMHom 𝑀) = ∅)
18		fvprc 6748	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
19	5, 18	syl5eq 2791	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝐴 = ∅)
20	19	fveq2d 6760	. . 3 ⊢ (¬ 𝑀 ∈ V → (Base‘𝐴) = (Base‘∅))
21	15, 17, 20	3eqtr4a 2805	. 2 ⊢ (¬ 𝑀 ∈ V → (𝑀 LMHom 𝑀) = (Base‘𝐴))
22	14, 21	pm2.61i 182	1 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881  ∅c0 4253  {csn 4558  {cpr 4560  {ctp 4562  ⟨cop 4564   × cxp 5578   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ∘_f cof 7509  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  Scalarcsca 16891   ·_𝑠 cvsca 16892   LMHom clmhm 20196  MEndocmend 40916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-lmhm 20199  df-mend 40917

This theorem is referenced by:  mendplusgfval  40926  mendmulrfval  40928  mendvscafval  40931  mendring  40933  mendlmod  40934  mendassa  40935