Theorem mdetleib 20761

Description: Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-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
mdetleib	⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))))

Distinct variable groups:   𝑥,𝑝,𝑀   𝑁,𝑝,𝑥   𝑅,𝑝,𝑥

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

Proof of Theorem mdetleib

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq 6911	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑥)𝑚𝑥) = ((𝑝‘𝑥)𝑀𝑥))
2	1	mpteq2dv 4968	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))
3	2	oveq2d 6921	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))
4	3	oveq2d 6921	. . . 4 ⊢ (𝑚 = 𝑀 → (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))) = (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))
5	4	mpteq2dv 4968	. . 3 ⊢ (𝑚 = 𝑀 → (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))) = (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))))
6	5	oveq2d 6921	. 2 ⊢ (𝑚 = 𝑀 → (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))))
7		mdetfval.d	. . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅)
8		mdetfval.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
9		mdetfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
10		mdetfval.p	. . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
11		mdetfval.y	. . 3 ⊢ 𝑌 = (ℤRHom‘𝑅)
12		mdetfval.s	. . 3 ⊢ 𝑆 = (pmSgn‘𝑁)
13		mdetfval.t	. . 3 ⊢ · = (.r‘𝑅)
14		mdetfval.u	. . 3 ⊢ 𝑈 = (mulGrp‘𝑅)
15	7, 8, 9, 10, 11, 12, 13, 14	mdetfval 20760	. 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
16		ovex 6937	. 2 ⊢ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) ∈ V
17	6, 15, 16	fvmpt 6529	1 ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166   ↦ cmpt 4952   ∘ ccom 5346  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  ._rcmulr 16306   Σ_g cgsu 16454  SymGrpcsymg 18147  pmSgncpsgn 18259  mulGrpcmgp 18843  ℤRHomczrh 20208   Mat cmat 20580   maDet cmdat 20758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-slot 16226  df-base 16228  df-mat 20581  df-mdet 20759

This theorem is referenced by:  mdetleib2  20762  m1detdiag  20771  mdetdiag  20773  mdetralt  20782  mdettpos  20785  chpmatval2  21008  mdetpmtr1  30434