Theorem maducoeval 22140

Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)

Hypotheses

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)

Assertion

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)))))

Distinct variable groups:   𝑘,𝑁,𝑙   𝑅,𝑘,𝑙   𝑘,𝑀,𝑙   𝑘,𝐼,𝑙   𝑘,𝐻,𝑙

Allowed substitution hints:   𝐴(𝑘,𝑙)   𝐵(𝑘,𝑙)   𝐷(𝑘,𝑙)   1 (𝑘,𝑙)   𝐽(𝑘,𝑙)   0 (𝑘,𝑙)

Proof of Theorem maducoeval

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

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 22139	. . 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 1133	. 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 1198	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑗 = 𝐻)
10	9	eqeq2d 2743	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑗 ↔ 𝑘 = 𝐻))
11		simp1l 1197	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑖 = 𝐼)
12	11	eqeq2d 2743	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑙 = 𝑖 ↔ 𝑙 = 𝐼))
13	12	ifbid 4551	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 ))
14	10, 13	ifbieq1d 4552	. . . . 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 7485	. . . 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 6895	. . 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 482	. 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 1137	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐼 ∈ 𝑁)
19		simp3 1138	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐻 ∈ 𝑁)
20		fvexd 6906	. 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 7559	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)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ifcif 4528  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17143  0_gc0g 17384  1_rcur 20003   Mat cmat 21906   maDet cmdat 22085   maAdju cmadu 22133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-nn 12212  df-slot 17114  df-ndx 17126  df-base 17144  df-mat 21907  df-madu 22135

This theorem is referenced by:  maducoeval2  22141  madugsum  22144  maducoevalmin1  22153