Theorem maducoevalmin1 20458

Description: The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)

Hypotheses

Ref	Expression
maducoevalmin1.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
maducoevalmin1.b	⊢ 𝐵 = (Base‘𝐴)
maducoevalmin1.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
maducoevalmin1.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)

Assertion

Ref	Expression
maducoevalmin1	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))

Proof of Theorem maducoevalmin1

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		maducoevalmin1.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		maducoevalmin1.d	. . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅)
3		maducoevalmin1.j	. . 3 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
4		maducoevalmin1.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
5		eqid 2622	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		eqid 2622	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	1, 2, 3, 4, 5, 6	maducoeval 20445	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗)))))
8		eqid 2622	. . . . . 6 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
9	1, 4, 8, 5, 6	minmar1val 20454	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐻 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
10	9	3com23 1271	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
11	10	eqcomd 2628	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) = (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))
12	11	fveq2d 6195	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗)))) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
13	7, 12	eqtrd 2656	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ifcif 4086  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  0_gc0g 16100  1_rcur 18501   Mat cmat 20213   maDet cmdat 20390   maAdju cmadu 20438   minMatR1 cminmar1 20439

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-mat 20214  df-madu 20440  df-minmar1 20441

This theorem is referenced by:  madjusmdetlem1  29893