Theorem mendplusgfval 37755

Description: Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)

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 37754	. . . . . . 7 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
4	2, 3	eqtr4i 2647	. . . . . 6 ⊢ 𝐵 = (𝑀 LMHom 𝑀)
5		mendplusgfval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
6		ofeq 6899	. . . . . . . . . 10 ⊢ ( + = (+g‘𝑀) → ∘𝑓 + = ∘𝑓 (+g‘𝑀))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ ∘𝑓 + = ∘𝑓 (+g‘𝑀)
8	7	oveqi 6663	. . . . . . . 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	mpt2eq3ia 6720	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 (+g‘𝑀)𝑦))
11		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
12		eqid 2622	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
13		eqid 2622	. . . . . 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 37753	. . . . 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	syl5eq 2668	. . . 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 6195	. . 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		fvex 6201	. . . . . 6 ⊢ (Base‘𝐴) ∈ V
18	2, 17	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
19	18, 18	mpt2ex 7247	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) ∈ V
20		eqid 2622	. . . . 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))⟩})
21	20	algaddg 37749	. . . 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))⟩})))
22	19, 21	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))⟩})))
23	16, 22	eqtr4d 2659	. 2 ⊢ (𝑀 ∈ V → (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)))
24		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (MEndo‘𝑀) = ∅)
25	1, 24	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝐴 = ∅)
26	25	fveq2d 6195	. . . 4 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = (+g‘∅))
27		df-plusg 15954	. . . . 5 ⊢ +g = Slot 2
28	27	str0 15911	. . . 4 ⊢ ∅ = (+g‘∅)
29	26, 28	syl6eqr 2674	. . 3 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = ∅)
30	25	fveq2d 6195	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (Base‘𝐴) = (Base‘∅))
31		base0 15912	. . . . . 6 ⊢ ∅ = (Base‘∅)
32	30, 2, 31	3eqtr4g 2681	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝐵 = ∅)
33		mpt2eq12 6715	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 ∘𝑓 + 𝑦)))
34	33	anidms 677	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 ∘𝑓 + 𝑦)))
35	32, 34	syl 17	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 ∘𝑓 + 𝑦)))
36		mpt20 6725	. . . 4 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 ∘𝑓 + 𝑦)) = ∅
37	35, 36	syl6eq 2672	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)) = ∅)
38	29, 37	eqtr4d 2659	. 2 ⊢ (¬ 𝑀 ∈ V → (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦)))
39	23, 38	pm2.61i 176	1 ⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘𝑓 + 𝑦))

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  {cpr 4179  {ctp 4181  ⟨cop 4183   × cxp 5112   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑓 cof 6895  2c2 11070  ndxcnx 15854  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945   LMHom clmhm 19019  MEndocmend 37745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-lmhm 19022  df-mend 37746

This theorem is referenced by:  mendplusg  37756