Theorem mendsca 41000

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.

Step	Hyp	Ref	Expression
1		fvex 6780	. . . . 5 ⊢ (Scalar‘𝑀) ∈ V
2		eqid 2738	. . . . . 6 ⊢ ({⟨(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	algsca 40992	. . . . 5 ⊢ ((Scalar‘𝑀) ∈ V → (Scalar‘𝑀) = (Scalar‘({⟨(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	. . . 4 ⊢ (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘({⟨(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		eqid 2738	. . . . . 6 ⊢ (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
6		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))
7		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))
8		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
9		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))
10	5, 6, 7, 8, 9	mendval 40994	. . . . 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))⟩}))
11	10	fveq2d 6771	. . . 4 ⊢ (𝑀 ∈ V → (Scalar‘(MEndo‘𝑀)) = (Scalar‘({⟨(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	4, 11	eqtr4d 2781	. . 3 ⊢ (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
13		scaid 17013	. . . . . 6 ⊢ Scalar = Slot (Scalar‘ndx)
14	13	str0 16878	. . . . 5 ⊢ ∅ = (Scalar‘∅)
15	14	eqcomi 2747	. . . 4 ⊢ (Scalar‘∅) = ∅
16		eqid 2738	. . . 4 ⊢ (MEndo‘𝑀) = (MEndo‘𝑀)
17	15, 16	fveqprc 16880	. . 3 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
18	12, 17	pm2.61i 182	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀))
19		mendsca.s	. 2 ⊢ 𝑆 = (Scalar‘𝑀)
20		mendsca.a	. . 3 ⊢ 𝐴 = (MEndo‘𝑀)
21	20	fveq2i 6770	. 2 ⊢ (Scalar‘𝐴) = (Scalar‘(MEndo‘𝑀))
22	18, 19, 21	3eqtr4i 2776	1 ⊢ 𝑆 = (Scalar‘𝐴)

Syntax hints:   = wceq 1539   ∈ wcel 2106  Vcvv 3430   ∪ cun 3885  ∅c0 4257  {csn 4562  {cpr 4564  {ctp 4566  ⟨cop 4568   × cxp 5583   ∘ ccom 5589  ‘cfv 6427  (class class class)co 7268   ∈ cmpo 7270   ∘_f cof 7522  ndxcnx 16882  Basecbs 16900  +_gcplusg 16950  ._rcmulr 16951  Scalarcsca 16953   ·_𝑠 cvsca 16954   LMHom clmhm 20269  MEndocmend 40986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-cnex 10915  ax-resscn 10916  ax-1cn 10917  ax-icn 10918  ax-addcl 10919  ax-addrcl 10920  ax-mulcl 10921  ax-mulrcl 10922  ax-mulcom 10923  ax-addass 10924  ax-mulass 10925  ax-distr 10926  ax-i2m1 10927  ax-1ne0 10928  ax-1rid 10929  ax-rnegex 10930  ax-rrecex 10931  ax-cnre 10932  ax-pre-lttri 10933  ax-pre-lttrn 10934  ax-pre-ltadd 10935  ax-pre-mulgt0 10936

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-om 7704  df-1st 7821  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-1o 8285  df-er 8486  df-en 8722  df-dom 8723  df-sdom 8724  df-fin 8725  df-pnf 10999  df-mnf 11000  df-xr 11001  df-ltxr 11002  df-le 11003  df-sub 11195  df-neg 11196  df-nn 11962  df-2 12024  df-3 12025  df-4 12026  df-5 12027  df-6 12028  df-n0 12222  df-z 12308  df-uz 12571  df-fz 13228  df-struct 16836  df-slot 16871  df-ndx 16883  df-base 16901  df-plusg 16963  df-mulr 16964  df-sca 16966  df-vsca 16967  df-mend 40987

This theorem is referenced by:  mendlmod  41004  mendassa  41005