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Mirrors > Home > MPE Home > Th. List > maducoevalmin1 | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
maducoevalmin1 | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))) |
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 2772 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | eqid 2772 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | maducoeval 20942 | . 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 2772 | . . . . . 6 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅) | |
9 | 1, 4, 8, 5, 6 | minmar1val 20951 | . . . . 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 1106 | . . . 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 2778 | . . 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 6497 | . 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 2808 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ifcif 4344 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 Basecbs 16329 0gc0g 16559 1rcur 18964 Mat cmat 20710 maDet cmdat 20887 maAdju cmadu 20935 minMatR1 cminmar1 20936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-slot 16333 df-base 16335 df-mat 20711 df-madu 20937 df-minmar1 20938 |
This theorem is referenced by: madjusmdetlem1 30691 |
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