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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 2651 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | eqid 2651 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | maducoeval 20493 | . 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 2651 | . . . . . 6 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅) | |
9 | 1, 4, 8, 5, 6 | minmar1val 20502 | . . . . 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 1291 | . . . 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 2657 | . . 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 6233 | . 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 2685 | 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 1054 = wceq 1523 ∈ wcel 2030 ifcif 4119 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 Basecbs 15904 0gc0g 16147 1rcur 18547 Mat cmat 20261 maDet cmdat 20438 maAdju cmadu 20486 minMatR1 cminmar1 20487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-slot 15908 df-base 15910 df-mat 20262 df-madu 20488 df-minmar1 20489 |
This theorem is referenced by: madjusmdetlem1 30021 |
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