Theorem mdetfval 22901

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 7429	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3		mdetfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
5	4	fveq2d 6889	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6		mdetfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8		simpr 490	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9		simpl 488	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
10	9	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
11	10	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12		mdetfval.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
13	11, 12	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
15	14	adantl 487	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.r‘𝑟) = (.r‘𝑅))
16		mdetfval.t	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
17	15, 16	eqtr4di 2814	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.r‘𝑟) = · )
18	8	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19		mdetfval.y	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤRHom‘𝑅)
20	18, 19	eqtr4di 2814	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
22	21	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23		mdetfval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (pmSgn‘𝑁)
24	22, 23	eqtr4di 2814	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
25	20, 24	coeq12d 5842	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌 ∘ 𝑆))
26	25	fveq1d 6887	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌 ∘ 𝑆)‘𝑝))
27		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
28	27	adantl 487	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29		mdetfval.u	. . . . . . . . . 10 ⊢ 𝑈 = (mulGrp‘𝑅)
30	28, 29	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
31	9	mpteq1d 5195	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))
32	30, 31	oveq12d 7438	. . . . . . . 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 7441	. . . . . . 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 5192	. . . . . 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 7438	. . . . 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 5192	. . . 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 22900	. . . 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 6899	. . . . 5 ⊢ 𝐵 ∈ V
39	38	mptex 7229	. . . 4 ⊢ (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) ∈ V
40	36, 37, 39	ovmpoa 7575	. . 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 7554	. . . . . 6 ⊢ Rel dom maDet
42	41	ovprc 7458	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
43		mpt0 6681	. . . . 5 ⊢ (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = ∅
44	42, 43	eqtr4di 2814	. . . 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 22723	. . . . . . . . . 10 ⊢ Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(.r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
46	45	reldmmpo 7554	. . . . . . . . 9 ⊢ Rel dom Mat
47	46	ovprc 7458	. . . . . . . 8 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
48	3, 47	eqtrid 2808	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
49	48	fveq2d 6889	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
50		base0 17392	. . . . . 6 ⊢ ∅ = (Base‘∅)
51	49, 6, 50	3eqtr4g 2821	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
52	51	mpteq1d 5195	. . . 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 2799	. . 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 184	. 2 ⊢ (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
55	1, 54	eqtri 2784	1 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ⟨cop 4590  ⟨cotp 4592   ↦ cmpt 5186   × cxp 5649   ∘ ccom 5655  ‘cfv 6538  (class class class)co 7420  Fincfn 8973   sSet csts 17341  ndxcnx 17371  Basecbs 17387  ._rcmulr 17429   Σ_g cgsu 17611  SymGrpcsymg 19583  pmSgncpsgn 19703  mulGrpcmgp 20360  ℤRHomczrh 21805   freeLMod cfrlm 22052   maMul cmmul 22705   Mat cmat 22722   maDet cmdat 22899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-nn 12336  df-slot 17360  df-ndx 17372  df-base 17388  df-mat 22723  df-mdet 22900

This theorem is used by:  mdetleib  22902  nfimdetndef  22904  mdetfval1  22905  mdet0pr  22907  mdetf  22910