Theorem mendsca 38436

Description: The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)

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 6388	. . . . 5 ⊢ (Scalar‘𝑀) ∈ V
2		eqid 2765	. . . . . 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 38428	. . . . 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 2765	. . . . . 6 ⊢ (𝑀 LMHom 𝑀) = (𝑀 LMHom 𝑀)
6		eqid 2765	. . . . . 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 2765	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ (𝑀 LMHom 𝑀), 𝑦 ∈ (𝑀 LMHom 𝑀) ↦ (𝑥 ∘ 𝑦))
8		eqid 2765	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
9		eqid 2765	. . . . . 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 38430	. . . . 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 6379	. . . 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 2802	. . 3 ⊢ (𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
13		df-sca 16230	. . . . 5 ⊢ Scalar = Slot 5
14	13	str0 16183	. . . 4 ⊢ ∅ = (Scalar‘∅)
15		fvprc 6368	. . . 4 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = ∅)
16		fvprc 6368	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
17	16	fveq2d 6379	. . . 4 ⊢ (¬ 𝑀 ∈ V → (Scalar‘(MEndo‘𝑀)) = (Scalar‘∅))
18	14, 15, 17	3eqtr4a 2825	. . 3 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀)))
19	12, 18	pm2.61i 176	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘(MEndo‘𝑀))
20		mendsca.s	. 2 ⊢ 𝑆 = (Scalar‘𝑀)
21		mendsca.a	. . 3 ⊢ 𝐴 = (MEndo‘𝑀)
22	21	fveq2i 6378	. 2 ⊢ (Scalar‘𝐴) = (Scalar‘(MEndo‘𝑀))
23	19, 20, 22	3eqtr4i 2797	1 ⊢ 𝑆 = (Scalar‘𝐴)

Syntax hints:  ¬ wn 3   = wceq 1652   ∈ wcel 2155  Vcvv 3350   ∪ cun 3730  ∅c0 4079  {csn 4334  {cpr 4336  {ctp 4338  ⟨cop 4340   × cxp 5275   ∘ ccom 5281  ‘cfv 6068  (class class class)co 6842   ↦ cmpt2 6844   ∘_𝑓 cof 7093  5c5 11330  ndxcnx 16127  Basecbs 16130  +_gcplusg 16214  ._rcmulr 16215  Scalarcsca 16217   ·_𝑠 cvsca 16218   LMHom clmhm 19291  MEndocmend 38422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-mend 38423

This theorem is referenced by:  mendlmod  38440  mendassa  38441