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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 2821 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | maducoeval 21242 | . 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 2821 | . . . . . 6 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅) | |
9 | 1, 4, 8, 5, 6 | minmar1val 21251 | . . . . 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 1122 | . . . 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 2827 | . . 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 6669 | . 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 2856 | 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 1083 = wceq 1533 ∈ wcel 2110 ifcif 4467 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 Basecbs 16477 0gc0g 16707 1rcur 19245 Mat cmat 21010 maDet cmdat 21187 maAdju cmadu 21235 minMatR1 cminmar1 21236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-slot 16481 df-base 16483 df-mat 21011 df-madu 21237 df-minmar1 21238 |
This theorem is referenced by: madjusmdetlem1 31087 |
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