Theorem mendsca 42725

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 6908	. . . . 5 ⊢ (Scalar‘𝑀) ∈ V
2		eqid 2725	. . . . . 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 42717	. . . . 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 2725	. . . . . 6 ⊢ (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
6		eqid 2725	. . . . . 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 2725	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))
8		eqid 2725	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
9		eqid 2725	. . . . . 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 42719	. . . . 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 6899	. . . 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 2768	. . 3 ⊢ (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
13		scaid 17297	. . . . . 6 ⊢ Scalar = Slot (Scalar‘ndx)
14	13	str0 17159	. . . . 5 ⊢ ∅ = (Scalar‘∅)
15	14	eqcomi 2734	. . . 4 ⊢ (Scalar‘∅) = ∅
16		eqid 2725	. . . 4 ⊢ (MEndo‘𝑀) = (MEndo‘𝑀)
17	15, 16	fveqprc 17161	. . 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 6898	. 2 ⊢ (Scalar‘𝐴) = (Scalar‘(MEndo‘𝑀))
22	18, 19, 21	3eqtr4i 2763	1 ⊢ 𝑆 = (Scalar‘𝐴)

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∪ cun 3942  ∅c0 4322  {csn 4630  {cpr 4632  {ctp 4634  ⟨cop 4636   × cxp 5676   ∘ ccom 5682  ‘cfv 6548  (class class class)co 7418   ∈ cmpo 7420   ∘_f cof 7682  ndxcnx 17163  Basecbs 17181  +_gcplusg 17234  ._rcmulr 17235  Scalarcsca 17237   ·_𝑠 cvsca 17238   LMHom clmhm 20913  MEndocmend 42711

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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6306  df-ord 6373  df-on 6374  df-lim 6375  df-suc 6376  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7374  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7684  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17182  df-plusg 17247  df-mulr 17248  df-sca 17250  df-vsca 17251  df-mend 42712

This theorem is referenced by:  mendlmod  42729  mendassa  42730

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