Theorem mdetfval 22471

Description: First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
mdetfval.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetfval.b	⊢ 𝐵 = (Base‘𝐴)
mdetfval.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
mdetfval.y	⊢ 𝑌 = (ℤRHom‘𝑅)
mdetfval.s	⊢ 𝑆 = (pmSgn‘𝑁)
mdetfval.t	⊢ · = (.r‘𝑅)
mdetfval.u	⊢ 𝑈 = (mulGrp‘𝑅)

Assertion

Ref	Expression
mdetfval	⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))

Distinct variable groups:   𝐵,𝑚   𝑚,𝑝,𝑥,𝑁   𝑃,𝑚   𝑅,𝑚,𝑝,𝑥   𝑆,𝑚   · ,𝑚   𝑈,𝑚   𝑚,𝑌

Allowed substitution hints:   𝐴(𝑥,𝑚,𝑝)   𝐵(𝑥,𝑝)   𝐷(𝑥,𝑚,𝑝)   𝑃(𝑥,𝑝)   𝑆(𝑥,𝑝)   · (𝑥,𝑝)   𝑈(𝑥,𝑝)   𝑌(𝑥,𝑝)

Proof of Theorem mdetfval

Dummy variables 𝑦 𝑧 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetfval.d	. 2 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		oveq12 7358	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3		mdetfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	2, 3	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
5	4	fveq2d 6826	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6		mdetfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8		simpr 484	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
10	9	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
11	10	fveq2d 6826	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12		mdetfval.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
13	11, 12	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
15	14	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.r‘𝑟) = (.r‘𝑅))
16		mdetfval.t	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
17	15, 16	eqtr4di 2782	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.r‘𝑟) = · )
18	8	fveq2d 6826	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19		mdetfval.y	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤRHom‘𝑅)
20	18, 19	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23		mdetfval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (pmSgn‘𝑁)
24	22, 23	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
25	20, 24	coeq12d 5807	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌 ∘ 𝑆))
26	25	fveq1d 6824	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌 ∘ 𝑆)‘𝑝))
27		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
28	27	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29		mdetfval.u	. . . . . . . . . 10 ⊢ 𝑈 = (mulGrp‘𝑅)
30	28, 29	eqtr4di 2782	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
31	9	mpteq1d 5182	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))
32	30, 31	oveq12d 7367	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))
33	17, 26, 32	oveq123d 7370	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))) = (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))
34	13, 33	mpteq12dv 5179	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))) = (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))
35	8, 34	oveq12d 7367	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
36	7, 35	mpteq12dv 5179	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
37		df-mdet 22470	. . . 4 ⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
38	6	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
39	38	mptex 7159	. . . 4 ⊢ (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) ∈ V
40	36, 37, 39	ovmpoa 7504	. . 3 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
41	37	reldmmpo 7483	. . . . . 6 ⊢ Rel dom maDet
42	41	ovprc 7387	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
43		mpt0 6624	. . . . 5 ⊢ (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = ∅
44	42, 43	eqtr4di 2782	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
45		df-mat 22293	. . . . . . . . . 10 ⊢ Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(.r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
46	45	reldmmpo 7483	. . . . . . . . 9 ⊢ Rel dom Mat
47	46	ovprc 7387	. . . . . . . 8 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
48	3, 47	eqtrid 2776	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
49	48	fveq2d 6826	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
50		base0 17125	. . . . . 6 ⊢ ∅ = (Base‘∅)
51	49, 6, 50	3eqtr4g 2789	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
52	51	mpteq1d 5182	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
53	44, 52	eqtr4d 2767	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
54	40, 53	pm2.61i 182	. 2 ⊢ (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
55	1, 54	eqtri 2752	1 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284  ⟨cop 4583  ⟨cotp 4585   ↦ cmpt 5173   × cxp 5617   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349  Fincfn 8872   sSet csts 17074  ndxcnx 17104  Basecbs 17120  ._rcmulr 17162   Σ_g cgsu 17344  SymGrpcsymg 19248  pmSgncpsgn 19368  mulGrpcmgp 20025  ℤRHomczrh 21406   freeLMod cfrlm 21653   maMul cmmul 22275   Mat cmat 22292   maDet cmdat 22469

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-1cn 11067  ax-addcl 11069

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-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-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-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-nn 12129  df-slot 17093  df-ndx 17105  df-base 17121  df-mat 22293  df-mdet 22470

This theorem is referenced by:  mdetleib  22472  nfimdetndef  22474  mdetfval1  22475  mdet0pr  22477  mdetf  22480