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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 2610 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | eqid 2610 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | maducoeval 20264 | . 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 2610 | . . . . . 6 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅) | |
9 | 1, 4, 8, 5, 6 | minmar1val 20273 | . . . . 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 1263 | . . . 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 2616 | . . 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 6107 | . 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 2644 | 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 1031 = wceq 1475 ∈ wcel 1977 ifcif 4036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 0gc0g 15923 1rcur 18324 Mat cmat 20032 maDet cmdat 20209 maAdju cmadu 20257 minMatR1 cminmar1 20258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-slot 15699 df-base 15700 df-mat 20033 df-madu 20259 df-minmar1 20260 |
This theorem is referenced by: madjusmdetlem1 29221 |
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