Theorem mendvscafval 43798

Description: Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)

Hypotheses

Ref	Expression
mendvscafval.a	⊢ 𝐴 = (MEndo‘𝑀)
mendvscafval.v	⊢ · = ( ·𝑠 ‘𝑀)
mendvscafval.b	⊢ 𝐵 = (Base‘𝐴)
mendvscafval.s	⊢ 𝑆 = (Scalar‘𝑀)
mendvscafval.k	⊢ 𝐾 = (Base‘𝑆)
mendvscafval.e	⊢ 𝐸 = (Base‘𝑀)

Assertion

Ref	Expression
mendvscafval	⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝑀,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑆(𝑥,𝑦)   · (𝑥,𝑦)   𝐸(𝑥,𝑦)

Proof of Theorem mendvscafval

Step	Hyp	Ref	Expression
1		mendvscafval.a	. . 3 ⊢ 𝐴 = (MEndo‘𝑀)
2	1	fveq2i 6882	. 2 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(MEndo‘𝑀))
3		mendvscafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
4	1	mendbas 43792	. . . . . . 7 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
5	3, 4	eqtr4i 2795	. . . . . 6 ⊢ 𝐵 = (𝑀 LMHom 𝑀)
6		eqid 2769	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))
7		eqid 2769	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
8		mendvscafval.s	. . . . . 6 ⊢ 𝑆 = (Scalar‘𝑀)
9		mendvscafval.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑆)
10		eqid 2769	. . . . . . 7 ⊢ 𝐵 = 𝐵
11		mendvscafval.e	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑀)
12	11	xpeq1i 5685	. . . . . . . 8 ⊢ (𝐸 × {𝑥}) = ((Base‘𝑀) × {𝑥})
13		eqid 2769	. . . . . . . 8 ⊢ 𝑦 = 𝑦
14		mendvscafval.v	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑀)
15		ofeq 7675	. . . . . . . . 9 ⊢ ( · = ( ·𝑠 ‘𝑀) → ∘f · = ∘f ( ·𝑠 ‘𝑀))
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ ∘f · = ∘f ( ·𝑠 ‘𝑀)
17	12, 13, 16	oveq123i 7422	. . . . . . 7 ⊢ ((𝐸 × {𝑥}) ∘f · 𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)
18	9, 10, 17	mpoeq123i 7484	. . . . . 6 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
19	5, 6, 7, 8, 18	mendval 43791	. . . . 5 ⊢ (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩}))
20	19	fveq2d 6883	. . . 4 ⊢ (𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩})))
21	9	fvexi 6893	. . . . . 6 ⊢ 𝐾 ∈ V
22	3	fvexi 6893	. . . . . 6 ⊢ 𝐵 ∈ V
23	21, 22	mpoex 8072	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) ∈ V
24		eqid 2769	. . . . . 6 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩})
25	24	algvsca 43790	. . . . 5 ⊢ ((𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩})))
26	23, 25	mp1i 14	. . . 4 ⊢ (𝑀 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))⟩})))
27	20, 26	eqtr4d 2807	. . 3 ⊢ (𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)))
28		fvprc 6871	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
29	28	fveq2d 6883	. . . . 5 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ( ·𝑠 ‘∅))
30		vscaid 17369	. . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
31	30	str0 17245	. . . . 5 ⊢ ∅ = ( ·𝑠 ‘∅)
32	29, 31	eqtr4di 2822	. . . 4 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ∅)
33		fvprc 6871	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = ∅)
34	8, 33	eqtrid 2816	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → 𝑆 = ∅)
35	34	fveq2d 6883	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → (Base‘𝑆) = (Base‘∅))
36		base0 17270	. . . . . . 7 ⊢ ∅ = (Base‘∅)
37	35, 9, 36	3eqtr4g 2829	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → 𝐾 = ∅)
38	37	orcd 886	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
39		0mpo0 7491	. . . . 5 ⊢ ((𝐾 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ∅)
40	38, 39	syl 18	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ∅)
41	32, 40	eqtr4d 2807	. . 3 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)))
42	27, 41	pm2.61i 184	. 2 ⊢ ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
43	2, 42	eqtri 2792	1 ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911  ∅c0 4294  {csn 4591  {cpr 4593  {ctp 4595  ⟨cop 4597   × cxp 5657   ∘ ccom 5663  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410   ∘_f cof 7670  ndxcnx 17249  Basecbs 17265  +_gcplusg 17306  ._rcmulr 17307  Scalarcsca 17309   ·_𝑠 cvsca 17310   LMHom clmhm 21114  MEndocmend 43783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-n0 12501  df-z 12588  df-uz 12859  df-fz 13532  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-lmhm 21117  df-mend 43784

This theorem is referenced by:  mendvsca  43799