Theorem mendplusgfval 43154

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

Hypotheses

Ref	Expression
mendplusgfval.a	⊢ 𝐴 = (MEndo‘𝑀)
mendplusgfval.b	⊢ 𝐵 = (Base‘𝐴)
mendplusgfval.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
mendplusgfval	⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))

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

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem mendplusgfval

Step	Hyp	Ref	Expression
1		mendplusgfval.a	. . . . 5 ⊢ 𝐴 = (MEndo‘𝑀)
2		mendplusgfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
3	1	mendbas 43153	. . . . . . 7 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
4	2, 3	eqtr4i 2755	. . . . . 6 ⊢ 𝐵 = (𝑀 LMHom 𝑀)
5		mendplusgfval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
6		ofeq 7616	. . . . . . . . . 10 ⊢ ( + = (+g‘𝑀) → ∘f + = ∘f (+g‘𝑀))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ ∘f + = ∘f (+g‘𝑀)
8	7	oveqi 7362	. . . . . . . 8 ⊢ (𝑥 ∘f + 𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦)
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∘f + 𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦))
10	9	mpoeq3ia 7427	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))
11		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
12		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
13		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)) = (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
14	4, 10, 11, 12, 13	mendval 43152	. . . . 5 ⊢ (𝑀 ∈ V → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}))
15	1, 14	eqtrid 2776	. . . 4 ⊢ (𝑀 ∈ V → 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}))
16	15	fveq2d 6826	. . 3 ⊢ (𝑀 ∈ V → (+g‘𝐴) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
17	2	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
18	17, 17	mpoex 8014	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) ∈ V
19		eqid 2729	. . . . 5 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})
20	19	algaddg 43148	. . . 4 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
21	18, 20	mp1i 13	. . 3 ⊢ (𝑀 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑀)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑀)), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))⟩})))
22	16, 21	eqtr4d 2767	. 2 ⊢ (𝑀 ∈ V → (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)))
23		fvprc 6814	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
24	1, 23	eqtrid 2776	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝐴 = ∅)
25	24	fveq2d 6826	. . . 4 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = (+g‘∅))
26		plusgid 17188	. . . . 5 ⊢ +g = Slot (+g‘ndx)
27	26	str0 17100	. . . 4 ⊢ ∅ = (+g‘∅)
28	25, 27	eqtr4di 2782	. . 3 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = ∅)
29	24	fveq2d 6826	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (Base‘𝐴) = (Base‘∅))
30		base0 17125	. . . . . 6 ⊢ ∅ = (Base‘∅)
31	29, 2, 30	3eqtr4g 2789	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝐵 = ∅)
32	31	olcd 874	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7432	. . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) = ∅)
34	32, 33	syl 17	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) = ∅)
35	28, 34	eqtr4d 2767	. 2 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)))
36	22, 35	pm2.61i 182	1 ⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901  ∅c0 4284  {csn 4577  {cpr 4579  {ctp 4581  ⟨cop 4583   × cxp 5617   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ∘_f cof 7611  ndxcnx 17104  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165   LMHom clmhm 20923  MEndocmend 43144

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-lmhm 20926  df-mend 43145

This theorem is referenced by:  mendplusg  43155