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Theorem mdetfval 22095

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 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (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 7420	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3		mdetfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	2, 3	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
5	4	fveq2d 6895	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6		mdetfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
7	5, 6	eqtr4di 2790	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8		simpr 485	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9		simpl 483	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
10	9	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
11	10	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12		mdetfval.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
13	11, 12	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (._r‘𝑟) = (._r‘𝑅))
15	14	adantl 482	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (._r‘𝑟) = (._r‘𝑅))
16		mdetfval.t	. . . . . . . . 9 ⊢ · = (._r‘𝑅)
17	15, 16	eqtr4di 2790	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (._r‘𝑟) = · )
18	8	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19		mdetfval.y	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤRHom‘𝑅)
20	18, 19	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
22	21	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23		mdetfval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (pmSgn‘𝑁)
24	22, 23	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
25	20, 24	coeq12d 5864	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌 ∘ 𝑆))
26	25	fveq1d 6893	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌 ∘ 𝑆)‘𝑝))
27		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
28	27	adantl 482	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29		mdetfval.u	. . . . . . . . . 10 ⊢ 𝑈 = (mulGrp‘𝑅)
30	28, 29	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
31	9	mpteq1d 5243	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))
32	30, 31	oveq12d 7429	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))
33	17, 26, 32	oveq123d 7432	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))) = (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))
34	13, 33	mpteq12dv 5239	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))) = (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))
35	8, 34	oveq12d 7429	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))) = (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
36	7, 35	mpteq12dv 5239	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
37		df-mdet 22094	. . . 4 ⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
38	6	fvexi 6905	. . . . 5 ⊢ 𝐵 ∈ V
39	38	mptex 7227	. . . 4 ⊢ (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) ∈ V
40	36, 37, 39	ovmpoa 7565	. . 3 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
41	37	reldmmpo 7545	. . . . . 6 ⊢ Rel dom maDet
42	41	ovprc 7449	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
43		mpt0 6692	. . . . 5 ⊢ (𝑚 ∈ ∅ ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = ∅
44	42, 43	eqtr4di 2790	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ ∅ ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
45		df-mat 21915	. . . . . . . . . 10 ⊢ Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(._r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
46	45	reldmmpo 7545	. . . . . . . . 9 ⊢ Rel dom Mat
47	46	ovprc 7449	. . . . . . . 8 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
48	3, 47	eqtrid 2784	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
49	48	fveq2d 6895	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
50		base0 17151	. . . . . 6 ⊢ ∅ = (Base‘∅)
51	49, 6, 50	3eqtr4g 2797	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
52	51	mpteq1d 5243	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = (𝑚 ∈ ∅ ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
53	44, 52	eqtr4d 2775	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
54	40, 53	pm2.61i 182	. 2 ⊢ (𝑁 maDet 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
55	1, 54	eqtri 2760	1 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 ⟨cop 4634 ⟨cotp 4636 ↦ cmpt 5231 × cxp 5674 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7411 Fincfn 8941 sSet csts 17098 ndxcnx 17128 Basecbs 17146 ._rcmulr 17200 Σ_g cgsu 17388 SymGrpcsymg 19236 pmSgncpsgn 19359 mulGrpcmgp 19989 ℤRHomczrh 21055 freeLMod cfrlm 21307 maMul cmmul 21892 Mat cmat 21914 maDet cmdat 22093

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-nn 12215 df-slot 17117 df-ndx 17129 df-base 17147 df-mat 21915 df-mdet 22094

This theorem is referenced by: mdetleib 22096 nfimdetndef 22098 mdetfval1 22099 mdet0pr 22101 mdetf 22104

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