Theorem mendvscafval 43614

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 6843	. 2 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(MEndo‘𝑀))
3		mendvscafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
4	1	mendbas 43608	. . . . . . 7 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
5	3, 4	eqtr4i 2762	. . . . . 6 ⊢ 𝐵 = (𝑀 LMHom 𝑀)
6		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))
7		eqid 2736	. . . . . 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 2736	. . . . . . 7 ⊢ 𝐵 = 𝐵
11		mendvscafval.e	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑀)
12	11	xpeq1i 5657	. . . . . . . 8 ⊢ (𝐸 × {𝑥}) = ((Base‘𝑀) × {𝑥})
13		eqid 2736	. . . . . . . 8 ⊢ 𝑦 = 𝑦
14		mendvscafval.v	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑀)
15		ofeq 7634	. . . . . . . . 9 ⊢ ( · = ( ·𝑠 ‘𝑀) → ∘f · = ∘f ( ·𝑠 ‘𝑀))
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ ∘f · = ∘f ( ·𝑠 ‘𝑀)
17	12, 13, 16	oveq123i 7381	. . . . . . 7 ⊢ ((𝐸 × {𝑥}) ∘f · 𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)
18	9, 10, 17	mpoeq123i 7443	. . . . . 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 43607	. . . . 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 6844	. . . 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 6854	. . . . . 6 ⊢ 𝐾 ∈ V
22	3	fvexi 6854	. . . . . 6 ⊢ 𝐵 ∈ V
23	21, 22	mpoex 8032	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) ∈ V
24		eqid 2736	. . . . . 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 43606	. . . . 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 2774	. . 3 ⊢ (𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)))
28		fvprc 6832	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
29	28	fveq2d 6844	. . . . 5 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ( ·𝑠 ‘∅))
30		vscaid 17283	. . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
31	30	str0 17159	. . . . 5 ⊢ ∅ = ( ·𝑠 ‘∅)
32	29, 31	eqtr4di 2789	. . . 4 ⊢ (¬ 𝑀 ∈ V → ( ·𝑠 ‘(MEndo‘𝑀)) = ∅)
33		fvprc 6832	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ V → (Scalar‘𝑀) = ∅)
34	8, 33	eqtrid 2783	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → 𝑆 = ∅)
35	34	fveq2d 6844	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → (Base‘𝑆) = (Base‘∅))
36		base0 17184	. . . . . . 7 ⊢ ∅ = (Base‘∅)
37	35, 9, 36	3eqtr4g 2796	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → 𝐾 = ∅)
38	37	orcd 874	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
39		0mpo0 7450	. . . . 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 2774	. . 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 2759	1 ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887  ∅c0 4273  {csn 4567  {cpr 4569  {ctp 4571  ⟨cop 4573   × cxp 5629   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ∘_f cof 7629  ndxcnx 17163  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224   LMHom clmhm 21014  MEndocmend 43599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-lmhm 21017  df-mend 43600

This theorem is referenced by:  mendvsca  43615