Theorem mendvscafval 43162

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 6829	. 2 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(MEndo‘𝑀))
3		mendvscafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
4	1	mendbas 43156	. . . . . . 7 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
5	3, 4	eqtr4i 2755	. . . . . 6 ⊢ 𝐵 = (𝑀 LMHom 𝑀)
6		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))
7		eqid 2729	. . . . . 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 2729	. . . . . . 7 ⊢ 𝐵 = 𝐵
11		mendvscafval.e	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑀)
12	11	xpeq1i 5649	. . . . . . . 8 ⊢ (𝐸 × {𝑥}) = ((Base‘𝑀) × {𝑥})
13		eqid 2729	. . . . . . . 8 ⊢ 𝑦 = 𝑦
14		mendvscafval.v	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑀)
15		ofeq 7620	. . . . . . . . 9 ⊢ ( · = ( ·𝑠 ‘𝑀) → ∘f · = ∘f ( ·𝑠 ‘𝑀))
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ ∘f · = ∘f ( ·𝑠 ‘𝑀)
17	12, 13, 16	oveq123i 7367	. . . . . . 7 ⊢ ((𝐸 × {𝑥}) ∘f · 𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)
18	9, 10, 17	mpoeq123i 7429	. . . . . 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 43155	. . . . 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 6830	. . . 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 6840	. . . . . 6 ⊢ 𝐾 ∈ V
22	3	fvexi 6840	. . . . . 6 ⊢ 𝐵 ∈ V
23	21, 22	mpoex 8021	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) ∈ V
24		eqid 2729	. . . . . 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 43154	. . . . 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 13	. . . 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 2767	. . 3 ⊢ (𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)))
28		fvprc 6818	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
29	28	fveq2d 6830	. . . . 5 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ( ·𝑠 ‘∅))
30		vscaid 17242	. . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
31	30	str0 17118	. . . . 5 ⊢ ∅ = ( ·𝑠 ‘∅)
32	29, 31	eqtr4di 2782	. . . 4 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ∅)
33		fvprc 6818	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = ∅)
34	8, 33	eqtrid 2776	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → 𝑆 = ∅)
35	34	fveq2d 6830	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → (Base‘𝑆) = (Base‘∅))
36		base0 17143	. . . . . . 7 ⊢ ∅ = (Base‘∅)
37	35, 9, 36	3eqtr4g 2789	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → 𝐾 = ∅)
38	37	orcd 873	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
39		0mpo0 7436	. . . . 5 ⊢ ((𝐾 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ∅)
40	38, 39	syl 17	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) = ∅)
41	32, 40	eqtr4d 2767	. . 3 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)))
42	27, 41	pm2.61i 182	. 2 ⊢ ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
43	2, 42	eqtri 2752	1 ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∪ cun 3903  ∅c0 4286  {csn 4579  {cpr 4581  {ctp 4583  ⟨cop 4585   × cxp 5621   ∘ ccom 5627  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355   ∘_f cof 7615  ndxcnx 17122  Basecbs 17138  +_gcplusg 17179  ._rcmulr 17180  Scalarcsca 17182   ·_𝑠 cvsca 17183   LMHom clmhm 20941  MEndocmend 43147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-lmhm 20944  df-mend 43148

This theorem is referenced by:  mendvsca  43163