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Mirrors > Home > MPE Home > Th. List > maducoeval | Structured version Visualization version GIF version |
Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.) |
Ref | Expression |
---|---|
madufval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
madufval.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
madufval.j | ⊢ 𝐽 = (𝑁 maAdju 𝑅) |
madufval.b | ⊢ 𝐵 = (Base‘𝐴) |
madufval.o | ⊢ 1 = (1r‘𝑅) |
madufval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
maducoeval | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | madufval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | madufval.d | . . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
3 | madufval.j | . . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑅) | |
4 | madufval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
5 | madufval.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
6 | madufval.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | maduval 22495 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))))) |
8 | 7 | 3ad2ant1 1130 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))))) |
9 | simp1r 1195 | . . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑗 = 𝐻) | |
10 | 9 | eqeq2d 2737 | . . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑗 ↔ 𝑘 = 𝐻)) |
11 | simp1l 1194 | . . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑖 = 𝐼) | |
12 | 11 | eqeq2d 2737 | . . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑙 = 𝑖 ↔ 𝑙 = 𝐼)) |
13 | 12 | ifbid 4546 | . . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 )) |
14 | 10, 13 | ifbieq1d 4547 | . . . . 5 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) |
15 | 14 | mpoeq3dva 7482 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) |
16 | 15 | fveq2d 6889 | . . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))) |
17 | 16 | adantl 481 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐻)) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))) |
18 | simp2 1134 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐼 ∈ 𝑁) | |
19 | simp3 1135 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐻 ∈ 𝑁) | |
20 | fvexd 6900 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V) | |
21 | 8, 17, 18, 19, 20 | ovmpod 7556 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ifcif 4523 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17153 0gc0g 17394 1rcur 20086 Mat cmat 22262 maDet cmdat 22441 maAdju cmadu 22489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-nn 12217 df-slot 17124 df-ndx 17136 df-base 17154 df-mat 22263 df-madu 22491 |
This theorem is referenced by: maducoeval2 22497 madugsum 22500 maducoevalmin1 22509 |
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